University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 

of 
Early  American  Mathematics  Books 


t 


THE 

MISSIONARY  ARITHMETIC  : 

OH 

ARITHMETIC  MADE  EASY^ 

m  A  NEW  METHOD : 

DESIGISTED   TO 

DLMINISH  THE  LABOR  OF  THE  TEACHER, 
INCREASE  THE  IMPROVEMENT  OF  THE  LEARNER. 

ACCOMMODATED   TO    ** 

THE  PRESENT  ERA  OF  BENEVOLENT  ENTERPRIZE, 

AND   ADAPTED   TO  THE  USE   OF 

LANCASTERIAN  AND  OTHER  SCHOOLS. 
BY  WILLIAM  R.  WEEKS. 

UTICA:* 

PUBLISHED  BY  MERRELL  &  HASTINGS, 

No,  40  Genesce-Sti'eet. 

William  VViUlams,  Pmiier^ 


JK^ortheni  IHstnct  of  J^e^v-York.  ss* 

BE  IT  REMEMBERED,  That  on  the  eighth  day  of  February,  in  the 
forty  sixth  year  of  the  Independence  of  the  United  States  of  America,  A.  D. 
182?,  Williara  R.  Weeks,  of  the  said  District,  has  deposited  in  this  office  the 
title  of  a  Book,  the  right  whereof  he  claims  t  s  author,  in  the  words  following, 
to  wit : 

"  The  Missionary  Arithmetic;  or  Arithmetic  made  easy,  in  a  new  method. 
Designed  to  diminish  the  labor  of  the  teacher,  and  increase  the  improvement 
of  the  learner  ;  accommodated  to  the  present  era  of  benevolent  enterprize, 
and  adapted  to  the  use  of  Lancasterian  £nd  other  schools.  By  William  R. 
Weeks." 

In  conformity  to  the  act  of  the  Congress  of  the  United  States,  entitled  "  An 
act  for  the  encouragement  of  learning,  by  securing  ihe  copies  of  maps,  charts, 
and  books,  to  the  authors,  and  proprietors  of  suth  copies,  during  the  time 
thereiQ  mentioned  ;"  and  also  to  an  act  entitled  *'  An  act  supplementary  to 
an  act  entitled  an  act  for  the  encouragement  of  learning  by  securing  the 
copies  of  maps,  charts,  and  books,  to  the  authors  ahd  proprietors  ot"  such 
copies,  during  the  time  therein  mentioned ;  and  extending  the  benefits  thereof 
to  the  arts  of  designing,  engraving  and  etching  historical  and  other  prints." 

RICHARD  R.  LANSING. 
Clerk  of  the  JVorthern  District  ofJS^exV'York, 


PREFACE, 


IN"  presenting  to  the  public  a  new  Arithmetic,  the  Author  ^v  ill 
floubtless  be  expected  to  oifer  soiae  reasons  for  such  a  publication. 
He  has  been  several  years  employed  in  the  instruction  of  schools,  and 
has  found  much  inconvenience  in  teaching:  arithmetic  from  the  books 
in  common  use,  and  much  interruption  to  the  oiher  business  of  the 
school,  from  the  study  of  it  in  the  common  way.  The  obscure 
Kianner  in  which  tlie  rules  are  usually  expressed,  and  the  want  of 
sufficient  illustration,  render  them  extremely  difficult  to  be 
comprehended  by  beginners.  This  discourages  the  learner  from 
committing  them  to  memory,  and  makes  him  impatient  to  go  forward 
and  attempt  to  perform  the  various  questions  which  are  inserted  tnr 
practice  in  the  rules.  In  doing  this,  he  meets  with  many  difficulties, 
owing  to  his  ignorance  of  the  rules,  and  is  perpetually  running  to 
the  instructor  for  assistance.  The  instructor  must  take  time  to  work 
out  his  questions  for  him,  or  spend  still  more  in  giving  explanations 
which  are  seldom  reriiembered.  And  if  he  does  not  stop  all  the 
other  business  of  the  school,  to  attend  thus  to  every  learner,  he  is 
complained  of  for  his  neglect,  and  the  learner  is  discouraged  from 
attempting  to  comprehend  what  appears  so  dark  and  perplexing. 
None  but  those  who  have  had  experience  in  teaching  schools,  can 
appreciate  the  trouble,  aad  vexation,  and  interruption,  which  con- 
tinually arise  from  this  source. 

To  furnish  an  Arithmetic,  and  to  point  out  a  mode  of  instruction, 
which  shall  remedy  tiiese  evils,  is  the  design  of  thi^  work.  The 
improvements  attempted  in  it,  are  the  following  : 

1.  The  rules  are  expressed  in  terms,  and  accompanied  by  expla- 
nations, more  easy  to  be  understood. 

2.  Under  every  rule,  one  question  or  more  is  performed  at  full 
length,  and  every  step  of  the  operation  explained  at  large,  that  the 
meaning  and  application  of  the  rule  may  be  clearly  seen. 

3.  Under  the  first  rules,  a  number  of  examples,  prepared  in  the 
Lancasterian  manner,  are  given,  for  classes  to  be  exercised  in  by 
the  help  of  a  monitor. 

4.  A  large  number  of  exercises  are  inserted,  which  consist  of  short 
and  easy  questions,  w  ith  answers  annexed,  to  be  performed  mentally, 
and  answered  extemporaneously,  to  a  monitor  j  designed  to  quicken 
the  attention  of  learners,  and  render  all  the  usual  operations  ii^ 
arithmetic  perfectly  familiar. 

5.  A  vset  of  questions  are  inserted,  on  the  nature  of  each  rule, 
without  answers,  in  the  manner  of  the  modern  improvements  in 
teaching  geography,  that  the  scholar  may  examine  the  rule  itself^ 
and  find  out  the  answers ;  intended  alsQ  for  the  use  of  classes. 


iv  Frejace. 

6.  The  ^eat  mass  of  questions  for  the  practice  of  learners,  are 
expressed  more  in  the  form  in  which  they  will  naturally  arise  in  the 
transaction  of  business,  anr^  are  set  down  by  themselves  in  the  second 
part  of  the  work,  without  answers  ;  and  after  a  few  olthe  hrst,  ihey 
are  not  arranged  in  the  order  of  the  rules  ;  so  that  the  learner,  in  order 
to  perform  them,  must  understand  his  rules,  and  pay  attention  to  the 
nature  of  the  questions  themselves.  The  numb-  r  of  them  is  also 
greater  than  usual,  that  the  learner  may  have,  in  going  through  the 
book,  abundant  ex<^ret-e  in  all  the  rules,  and  in  all  -iorts  of  questions 
that  will  be  likely  to  arise  in  the  transaction^!  of  active  life. 

7.  In  forming  the  questions  for  practice,  a  large  number  of  useful 
and  interesting  facts  are  embraced,  which  will  not  only  serve  the 
purpose  ofextrcising  the  learner  in  the  rules,  but  willConve}  to  him 
much  important  information  respecting  the  great  enterprizes  of 
Christian  benevolence  which  distinguish  the  present  age,  and  render 
this  work  a  useful  auxiliary,  in  training  the  rising  generation  to  es- 
teem the  privilege,  and  practise  the  duty  of  doing  good. 

8.  The  rules  for  extracting  the  roots,  are  expressed  in  a  new  f©rm, 
more  easy  to  be  understood  and  remembered. 

9.  Under  the  head  of  Mensuration,  easy  rules  are  given  for  finding 
the  content  of  the  various  solids,  the  capacity  of  difiierent  vessels, 
the  measurement  of  heigiits  and  distances,  and  the  surveying  of  land, 
as  far  as  is  necessary  for  the  common  purposes  of  the  tarmer,  without 
the  aid  of  mathematical  instruments. 

1 0.  The  whoile  is  adapted  to  the  use  of  schools,  in  such  a  maimer 
that  all  classes  of  learners  may  receive  the  requisite  attention  and 
instruction,  with  very  little  trouble  to  the  instructor,  and  very  little 
interruption  to  the  other  business  of  the  school. 

These  iuiprovemeuts  have  been  the  result  of  several  years'  atten- 
tion to  the  subject ;  and  most  of  them  have  had  the  test  of  experi- 
ence, in  schools  under  the  direction  of  the  Author,  long  enough  t© 
demonstrate  their  utility. 

For  the  valuable  hints  with  which  the  Author  hai  been  obligingly 
furnished  by  various  literary  ^tntlemen,  he  woiild  begtfeem  to  accept 
his  thanks.  He  is  particularly  indebted  to  the  Rev.  Joseph  Emerson, 
of  the  Byfield  Seminary ;  Mr.  Professor  S  rong,  of  Hamilton  College  ; 
Mr.  John  Randel,  Jun  of  Albany,  Surveyor;  Mr.  Luther  Jackson, 
of  New -York,  Teacher;  and  the  publiiatiotts  of  Mr.  Joseph  Lan- 
caster. 

As  some  errors  are  almost  inseparable  from  a  first  impression  of  a 
work  of  this  nature,  those  who  may  discover  any,  will  confer  a  favor 
on  the  Aiithor,  by  transmitting  to  him,  or  to  the  publisher-,  a  state- 
ment of  them,  that  they  may  be  corr^icted  in  a  subsequent  editioa. 

Paris,  February,  1822, 


ARITHMETIC  MADE  EASYo 

PART  T. 


Arithmetic  is  tiie  science  of  Numbers,  and  the  art  of  using 
them. 

JVotation  teaches  how  to  express  any  number  by  the  fol- 
lowing characters,  called  figures;   1,  2,  3,  4,  5,  6,  7,  8,  9,  0. 

JWimeration  teaches  how  to  read,  in  the  proper  words, 
any  number  expressed  by  these  figures. 

When  the  figures  stand  one  by  one,  their  value  is  as  fol- 
lows :  1,  one,  2,  two,  3,  three,  4,  four,  5,  five,  6,  six,  7,  seven, 
8,  eight,  9,  nine,  0,  nought ;  which  is  called  tiieir  simple  val- 
ue. Besides  their  simple  value,  they  have  another,  when  two 
or  more  of  them  are  joined  together,  which  depends  on  the 
place  in  which  they  stand,  and  which  may  be  called  their  lo- 
cal value. 

The  places  are  counted  from  right  to  left,  as  follows  :  first, 
units;  second,  tens;  third,  hundreds;  fourth,  thousands; 
fifth,  tens  of  thousands ;  sixth,  hundreds  of  thousands ;  se- 
venth, millions;  eighth,  tens  of  millions  ;  ninth,  hundreds 
of  millions  :  tenth,  thousands  of  millions  ;  and  so  on.  Thus, 
3,  standing  alone,  is  in  the  first  place,  and  denotes  three  units, 
or  three.  With  a  cypher  at  the  right  hand  of  it,  thus,  30, 
the  3,  standing  in  the  second  place,'  denotes  three  tens,  or 
thirty.  With  two  cyyjhers,  thus,  300,  it  stands  in  tlie  third 
place,  and  denotes  three  hundred.  With  three  cyphers,  thus, 
5000,  it  stands  in  the  fourth  place,  and  denotes" three  thou- 
sand. With  four  cyphers,  thus,  3  000,  it  stands  in  the  fifth 
place,  and  denotes  three  tens  of  thousands,  or  thirty  thou- 
sand. AVith  five  cyphers,  thus,  300000,  it  stands  in  the  sixth 
place,  and  denotes  three  hundreds  of  thousands,  or  three 
hundred  thousand.  With  six  cyphers,  thus,  3000000,  it 
stands  in  the  seventh  place,  and  denotes  three  millions.  With 
seven  cyphers,  thus,  30000000,  it  stands  in  the  eighth  place, 
and  denotes  diree  tens  of  millions,  or  thirty  millions.  As 
in  the  following  table : 

AS 


JS^umeration, 


r^  ^  a  H  H  ffi 


5 

3 

3  0 

3  0  0 

3 

0  0  0 

3  0 

0  0  0 

3  0  0 

0  0  0 

0  0  0 

0  0  0 

0  0  0 

0  0  0 

Three. 

Thirty. 

Three  hundred. 

Three  thousand. 

Thirty  thousand. 

Three  hundred  thousand. 

Three  millions. 

Thirty  millions. 

From  which  it  is  plain,  that  every  remove  a  figure  makefe 
from  the  right  hand  towards  the  left,  increases  its  value  ten- 
fold ;  its  value  in  the  column  of  tens,  being  ten  times  as  much 
as  in  the  column  of  units  ;  and  in  the  column  of  hundreds, 
ten  times  as  much  as  in  the  column  of  tens,  and  so  on.  The 
cypher  has  no  value  of  its  own,  but  only  serves  to  show^  the 
local  value  of  other  figures  to  which  it  is  annexed. 

8o  also,  when  other  figures  are  put  in  the  places  of  these 
cyphers,  each  figure  has  a  value  according  to  the  place  in 
v.hich  it  stands.  Thus,  in  21,  the  1,  standing  in  the  first 
place,  denotes  1  unit,  or  one,  and  the  2,  standing  in  the  se- 
cond place,  denotes  2  tens,  or  twenty ;  and  taken  together, 
they  are  to  be  read,  twenty-one.  In  32 1,  the  1,  standing  in 
the  first  place,  is  one  unit,  or  one ;  the  2,  standing  in  the  se- 
cond place,  is  two  tens,  or  twenty ;  and  the  3,  standing  in  t]w. 
third  place,  is  3  hundreds,  or  three  hundred;  and  taken  to- 
gether, they  are  to  be  read,  three  hundred  and  twenty  one. 
See  the  folio wino;  table : 


o 


0    k^=    § 

U    tr^    1  One. 

2  I  Twenty-one. 

3  2  1  Three  hundred  and  twenty -one. 

4  3  2  1  Four  thousand,  three  hundred  &  twenty -ene. 

5  4  3  2  1  Fifty-four  thousand,  3  hundred  and  21. 

6  5  4  3  2  1  Six  hundred  &  54  thousand,  3  bund.  &  21, 

7  6  5  4  3  2  1  Seven  million,  654  thousand,  3  hund.  &  21, 

8  7  6  5  4  3-21  Eighty-seven  million,  654  thous.  321. 

987654321  Nine  hund.  ^  87  million,  6  Imnd.  &  54  thoTis.  S2I, 


JSTumeration^ 


QUESTIONS    ON  THE  FOREGOING. 


How  much  does  a  ftgui'e  increase  its 
value,  by  every  remove  from  right 
to  left  f 

How  do  you  read  the  figures  1,  iJ,  3, 
4,  when  placed  so  that  1  shall  stand 
in  the  first  place,  2  in  tlie  second, 
3  in  the  third,  and  4  i  *  the  fourtli  ? 

How  do  you  read  the  figures,  !♦  2,  3, 
4,  5,  6,  standing  1  in  the  first  place, 
2  in  the  second,  and  so  on  ? 

How,  tlie  figures,  1,  2,  3,  4  5,  6,  7, 
after  the  same  an-angement  ? 

How  do  you  read  the  same  figures, 
when  7  stands  in  the  first  place,  6 
in  the  second,  5  in  the  third,  and 
so  on  ? 


What  is  Aritlfmetic  ?. 
What  does  Notation  teach  ? 
What  does  Numeration  teach  ? 
'What  is  meant  by  the  simple  value 

of  any  figure  ? 
What  by  its  local  value  ? 
Where   do  you  begin   to   count  the 

places  of  figures,  in  N^umeration? 
What  is  the  name  of  tlie  first  place  P 

the  second  f  third  ?  fourth  ?  fifth  ? 

sixth  ?    seventh  ?   eighth  ?  ninth  f 

tenth  ? 
What   is   the  value  of  tlie  figure    3, 

standing  in  the  third  place,   with 

cyphers  at  tlie  right  hand  of  it  ? 
What,  in  the  second  place  P  the  fifth  ? 

fourth  ?  eighth  ?  seventh  ?   sixth  ? 

J\''ote.  To  use  the  following  exercise,  let  the  class  be  seated  with  their 
slates,  but  without  any  books.  Let  the  Monitor  take  a  book,  and  read  the 
Avords,  twenty-one  ;  and  let  every  boy  write  down,  at  the  top  of  his  slate,  the 
iigures  which  express  that  number.  Then  let  the  Monitor  examine  all  the 
slates,  and  if  any  one  has  not  written  it  right,  show  him  how  to  do  it.  i  hta 
let  him  read  the  words  thirty-two^  and  let  them  write  it  as  before  ;  and  soon, 
placing  the  numbers  under  each  other,  as  they  stand  in  the  table,  units  under 
units,  tens  under  tensj  &c.  When  the  slates  are  filled,  let  them  read  their 
iigures  in  words,  the  first  boy  reading  the  first  number,  and  the  second  the 
itext,  and  so  on  ;  the  Monitor  looking  over  the  words  ik  the  table,  to  see  if 
ihey  read  them  right. 

Exercise  1. 
21  Twenty- one. 
32  Thirty-two. 
524  Five  hundred  and  twenty- four. 
78  Seventy- eight. 
169  Oae  hundred  and  sixty -nine. 
436  Four  hundred  and  thirty- six. 
1234  One  thousand,  two  hundred  and  thirty-four. 
3451   Three  thousand,  four  hundred  and  fifty- one« 

89  Eighty-nine. 
643  Six  hundred  and  forty -three. 
4326  Four  thousand,  three  hundred  and  twenty  six. 
235  Two  hundred  and  thirty-five. 
6478  Six  thousand,  four  hundred  and  seventy- eight. 
£1564  Twenty-one  thousand,  five  hundred  &,  sixty -four. 
987  Nine  hundred  aYid  eighty  seven, 
99  Ninety-nine. 
34567  Thirty  four  thousand,  five  hundred  and  67. 
2348673  Two  million,  348  thousand,  6  hundred  and  73. 
6542  Six  thousand,  five  hundred  and  forty  two. 
129834  One  hundred  and  29  thousand,  8  huadred  and  34. 


•  J^meration* 

J^'oU,  When  ♦he  tlass  are  sufficiently  practised  in  this  exereisei  let  then* 
take  exei-cis'S  t  and  3  After  they  have  had  a  little  practice  in  them,  let 
questions  be  given  ftw.n  all  promiscuoasly.  And  lot  no  boy  proceed,  to  the 
nrxt  rule,  till  he  is  tboroughlv  acquainted  with  ail  that  goew  before,  and  can 
write  (lowji  correctly,  from  the  mouth  of  the  Instructor,  any  number  he  shall 
ilictatc,  and  read  correctly  any  number  he  shall  write. 

Exercise  2. 
12  Twelve. 
201  Two  hundred  and  one. 
2020  Two  thousand  and  twenty. 
2200   Two  thousand,  two  hundred. 
2002  Two  thousand  and  two. 
S0303  Thirty  thousand,  three  hundred  and  three. 
303030  Three  hundred  and  three  thousand  and  thirty. 
330303  Three  hundred  &  30  thousand,  3  hundred  &  3. 
3033003  Three  million,  thirty -three  thousand  and  three. 

31091  Thirty  one  thousand  and  ninety  one. 
4040  vO  Four  hundred  and  four  thousand  and  forty. 
4014  Four  thousand  and  fourteen. 
4004041  Four  million,  four  thousand  and  forty-one. 
15115   Fifteen  thousand,  one  hundred  and  fifteen. 
505^-5  Fifty  thousand,  five  hundred  and  five. 
6606060  Six  million,  six  hundred  and  6  thousandand  60. 

60006  Sixty  thousand  and  six. 
9901019  Nine  million,  nine  hundred  and  1  thousand  and  19. 
109090  One  hundred  and  nine  thousand  and  ninety, 
rroori  Seven  hundred  and  seventy  thousand  and  71. 

Exercise  3. 
101   One  hundred  and  one. 
110  One  hundred  and  ten. 
1001  One  thousand  and  one. 

1010  One  thousand  and  ten. 

Ill  0  One  thousand,  one  hundred. 

1011  One  thousand  and  eleven. 
11001  Fiieven  thousand  and  one. 
10100  Ten  thousand,  one  hundred. 

10010  Ten  thousand  and  ten.- 
100001  One  hundred  thousand  and  one. 

101100  One  hundred  and  one  thousand,  one  hundred. 

11010  Eleven  thousand  and  ien. 

10011  Ten  thousand  and  eleven. 

100101  One  hundred  thousand,  one  hundred  and  one; 

1101 1  Eleven  thousand  and  eleven. 

10111  Ten  thousand,  one  hundred  and  eleven. 


I 


Bxplanation  of  Characters. 


101011  One  hundred  and  one  thousand  and  eleven. 

1 1101  Eleven  thousand,  one  hundred  and  one. 

1100001  One  million,  one  hundred  thousand  and  one. 

1101101  One  million,  one  hundred  and  one  thousand,  101. 


EXPLANATION  OF  CHAltACTERS. 

Two  parallel  horizontal  lines  signify  equality,  as  100  centt 
=  1  dol.  that  is,  100  cents  equal  1  dollar. 
Ir}-  A  cross,   made  by  a  horizontal  line  and  another  perpen- 
dicular to  it,  is  the  sign  of  addition,  as  2+4  =s6,  that  is, 
the  sum  of  2  and  4,  is  equal  to  6. 

A  horizontal  line,  is  the  sign  of  subtraction,  and  shovi^s 
that  the  number  which  stands  after  it  is  to  be  taken 
from  the  number  which  stands  before  it,  as  6 — 2=4, 
that  is,  6  diminished  by  £,  is  equal  to  4. 

A  cross,  like  the  Roman  letter  X,  is  the  sign  of  multipli- 
cation, as  3x6z=  1 8,  that  is  3  times  6  is  equal  to  18. 
t-^  A  horizontal  line,  with  a  point  above  and  below  it,  is  the 
sign  of  division,  and  shows  that  the  number  wh'ch 
stands  before  it  is  to  be  divided  by  that  which  stands 
after  it,  as  24 —6 =4,  that  is,  24  divided'by  6,  is  equal 
to  4. 

:  :     :  Points  standing  one  above  another  like  colons,  arc 
used  to  signify    proportion.     That  is,  when  four  num- 
bers are  placed  in  succession,  with  one  colon  between 
the  first  and  second,  two  colons   between  the  second 
and  third,  and  one  colon  between  the  third  and  fourth, 
they  signify  that  the  first  number  has  the  same  propor- 
tion to  the  second,  that  the  third  has  to  the  fourth  ;  thus, 
2  :  4  :  :  8  :  16,  that  y$,  as  2  is  to  4,  so  is  8  to  16. 
••    Two  points,  standing  beside  each  other,  are  used  in  this 
work  to  separate  different  denominations  ;  as,  ^  2  •• 
6  ••  8.  that  is,  2  pounds,  6  shilling:^,  and  8  pence. 
•     A  single  point  is  used  in  decimal  fractions,  to  separate 
the  whole  numbers  from  the  decimal  parts,  as  8*5,  that 
is  8  and  5  tenths.     It  is  also  used  to  separate  dollars 
from  cents  and  mills,  because  cents  and  mills  are  de- 
cimal pans  of  a  dollar;  as,  SS'65r,  that  is,  3  dollars, 
65  cents  and  7  mills. 
J     One  number  written  over  another,  with  a  line  between, 
is  called  a  vulgar  fraction  ;  as  J  one  half,  i  one  third, 
J  three  fourthi. 


10  Addition. 

JVote  7velL  The  learner  should  be  careful  not  to  make  any  of  these  mai^ts 
«pon  his  slate  or  paper,  for  any  other  purpose,  or  with  any  other  meaning, 
flian  is  here  directed;   and  to  make  no  unnecessary  marks  whatever. 

Questions  on  the  foregoing. 

What  is  tlie  sign  of  equality  ?  of  ad-  For  what  is  a  single  point  used  ? 

dition  ?      subtraction  ?      multipli-  Why  is  it  used  for  tlie  laf  ter  ? 

cation  ?  division  ?  proportion  ?  flow  are  vulgar  fractious  written  ? 

"What  marks  are  used  to  separate  What  cant  on  should  the  leai'uer  oV 

different  denominations  ?  serve  about  mai-ks  ? 


FUNDAMENTAL  RULES. 

There  are  four  rules  which  are  called  the  Fundamental 
Rules,  because  all  operations  in  arithmetic  are  perforjued  bj 
the  use  of  them.  Thej  are  Addition,  Subtraction,  Multipli- 
eation,  and  Division. 

ADDITION, 

Is  putting  together  two  or  more  numba's,  so  as  to  find  their 
total  amount,  which  is  called  their  sum. 

It  is  calleti  Simple.  Addition,  when  the  numbers  to  be  put 
together  are  all  of  the  same  denomination. 

Rule. 

1.  Write  down  the  several  numbers  under  each  other,  s« 
that  units  shall  stand  under  units,  tens  under  tens,  &c. 

2.  Draw  a  line  under  the  lowest  number,  to  separate  the 
^iven  numbers  from  their  sum,  when  it  shall  be  found. 

3.  Take  the  right  hand  column,  or  row  of  units,  begin  at  the 
bottom,  and  add  up.  If  the  sum  of  that  column  is  les»  than 
ten,  that  is,  if  it  is  but  one  figure,  it  is  units,  and  you  must 
set  it  down  under  the  column  of  units,  and  proceed  to  the 
next  column.  If  it  is  ten  or  more,  that  is,  if  it  is  more  than 
one  figure,  set  down  the  right  hand  figure,  which  is  a  unit, 
under  the  column  "of  units,  and  carry  the  rest,  which  will  be 
tens,  and  add  them  to  the  column  of  tens. 

4.  Add  up  the  column  of  tens,  and  when  you  have  found 
the  sum,  if  it  is  but  one  figure,  it  is  tens,  and  you  must  set  it 
down  under  the  column  of  tens ;  but  if  it  is  more  than  one 
figure,  the  right  hand  one  is  tens,  and  must  be  set  down  under 
the  column  of  tens,  and  the  rest  will  be  hundreds,  and  must 
be  carried  and  added  to  the  column  of  hundreds. 

5.  Proceed  in  like  manner  through  all  the  columns  to  the 
last,  where  you  must  set  down  the  whole  amount  of  that 
•olumu. 


I 


Mdition*  li 


Proof. 

Begin  at  the  top,  and  add  downwards,  and  if  the  total  is 
the  same  as  the  first  total,  the  work  is  probably  right. 
Example. 
Find  the  sum  of  S21,  43b,  372,  and  647. 
First,  I  write  down  the  numbers  under  each  other,  so  that 
units  stand  under  units,  tens  under  tens,  &c. ;  then  I  draw  a 
line  under,  and  add  as  follows  : 
321         I  begin  at  the  right  hand  column,  at  the  bottom,  and 
436     say,  7  and  2  is  9,  and  §  is  15,  and  1  is  16.     llie  right 
372     hand  figure  6,  being  units,  I  set  down  under  the  co- 
647    lumn  of  units ;  and  the  other  1 ,  being  a  ten,  I  carry  to 

the  column  of  tens,  and  say,  1  to  4  is  5,  and  7  is  12, 

1776     and  3  is  15,  and  2  is  17.     This  being  the  sum  of  the 
column  of  tens,  is  17  tens,  or  one  hundred  and  seventy. 
The  right  hand  figure  7,  being  tens,  and  denoting  7  tens  or 
70,  I  set  down  under  the  column  of  tens,  and  cairy  the  other 
1,  being  a  hundred,  to  the  column  of  hundreds,  and  say,  1   to 
6  is  7,  and  3  is  10,  and  4  is  14,  and  3  is  17.     This  being  the 
sum  of  the  column  of  hundreds,  is  17  hundreds,  or  one  thou- 
sand seven  hundred.     The  7,  denoting  seven  hundreds,  I  set 
down  under  the  column  of  hundreds  ;  and  the  i,  denoting  one 
thousand,  I  set  down  in  the  place  of  thousands,  there  being 
no  column  of  thousands  to  which  to  carry  it.  And  the  answer 
is,  one  thousand,  seven  hundred  and  seventy-six. 
Proof. 
321         I  begin  at  the  right  hand  column  at  the  top,  and  say, 
436     1  and  6  is  7,  and  2  is  9,  and  7  is  16.     Set  down  6  un- 
372     der  the  T,  and  carry  1  to  the  next  column.     1  and  2  is 
647    S,  and  3  is  6,  and  7  is  13,  and  4  is  17.     Set  down  7 

under  the  4,  and  carry  1  to  the  next.     1  and  3  is  4, 

1776     and  4  is  8,  and  3  is  11,  and  6  is  17.     Set  downaz. 
And  the  total  is  1776,  the  same  as  before  ;  so  1  con- 
clude the  work  is  right. 

JYote  1  o  use  the  following  example^,  let  a  c^ass  be  seated  with  their  slates, 
and  let  thfc  Monitor  take  the  book  ar»d  read  the  first  number,  and  let  it  be 
taken  down  and  examined.  Then  let  him  read  the  second  number,  and  see 
that  that  is  taken  down  correcth,  and  placed  under  the  first,  so  that  units 
shui-  stand  under  units,  tens  under  tens,  &c  W  hen  aU  the  numbers  are 
correctly  tak<  n  down,  and  a  line  drawn  under,  let  him  read  the  work  as  it  is 
set  down  under  the  question,  repealing  it  slowly  and  distinctly.  Whilu  he 
readfej,  let  each  boy  fijlow  him  up  tlie  column,  pointing  to  each  figiu'e  as  the 
monitor  names  it,  and  taking  notice  of  th':;  amount  which  it  makes  ;  and 
when  the  Monitor  tells  what  figure  to  set  down  at  the  bottom  of  the  column, 
\*ti  each  boy  set  it  dowxi ;  and  so  on,  till  the  -vrhole  is  finished.    And  when 


12  Jtdditisn. 

#ie  Monitor  reads  the  apiount,  let  each  boj  read  it  after  him,  from  his  slate. 
When  this  is  done,  let  thai  work  bo  rubbed  out,  and  another  example  per- 
formed in  the  same  inrinner  When  all  the  examples  havejaeen  several  times 
repeated  in  ihis  Avay,  let  th^  Monitor  vary  the  process,  in  this  manner;  let 
Mm  nnme  the  fiist  and  second  fij^^m  es,  and  instead  of  reading  from  his  hook 
iB^hat  th'7  amount  to,  iet  the  firat  bo}  tell  ,  tlien  let  the  Monitor  repeat  the 
amount,  and  n  nie  the  next  fiij;uie,  and  the  second  b(,y  tell  the'amouut,  and 
BO  Oi),  till  the  whole  is  finished.  If  one  boy  tells  wrong,  let  it  be  put  to  the 
mext,  and  if  no  oiie  cau  teU  right,  let  the  Monitor  ttll. 

JSTo,  1. 

27935  Work,  Take  the  right  hand  column,  and  begin  at 
3963  the  bottom.  7  and  9  is  16,  and  3  is  19,  and  5  is  24  ; 
8679  set  down  4  under  the  7,  and  carry  9.  to  the  next. 
143. 7  Second  column.  2  and  2  is  4,  and  7  is  11,  and  6  is 
— — —  17,  and  3  is  20 ;  set  down  0  under  the  2,  and  carrj 
54904     2  to  the  next. 

Tiiivd  cUumn,  2  and  S  is  5,  and  6  is  11,  and  9  i» 
SO,  and  9  is  29 ;  set  down  9  under  the  3,  and  carry  2  to  the 
next 

Fourth  column.  2  and  4  is  6,  and  8  is  «4,  and  3  is  17,  and 
7  is  24  ;  set  down  4  under  the  4,  and  carry  2  to  the  next. 
Fifth  column.  2  and     is  3,  and  2  is  5  ;  set  down  5. 
Total,  in   figures,    4904;  in  words,  fifty-four  thousand, 
nine  hundred  and  four. 

JVo.  2. 

12345  Work,  first  column.  5  and  1  is  6,  and  6  is  12,  and 

678  >  6  is   15,  and  i^  is  27,  and  5  is  32;  set  down  v,  and 

S2356  cany  3. 

7890  Second  column.  3  and  4  is  7,  and  9  is  16,  and  5  is 
13456  21,  and  9  U  30,  and  5  is  35,  and  8  is  43,  and  4  is  47  ; 

7891  set  down  7,  and  carry  i. 

2845         Third  column.  4  and  3  is  7,  and  8  is  15,  and  4  is 

1 9,  arid  8  is  27,  and  3  is  30,  and  7  is  37,  and  3  is  40  ; 

83072     set  down  0,  and  carry  4. 

Fourth  column.  4  and  2  is  G,  and  7  is  13,  and  3  is 
16,  and  7  is  23,  and  2  is  25,  and  6  is  31,  and  2  is  33 ;  set 
down  3,  and  carry  3. 

Fifth  column.  3  and  1  is  4,  and  3  is  7,  and  1  is  8  ;  set 
down  8. 

Total,  in  figures,  83072;  in  words,  eighty -three  thousand, 
ftftd  seventy -tw». 


"i, 


Addition. 


15 


b.  5, 

56784 
90235 
45676 
81237 
45988 
76549 
32131 
45802 
72343 
56784 
92345 


Work  ;  first  column.  5  and  4  is  9,  and  S  is  12, 
and  2  is  14,  and  1  is  15,  and  9  is  24,  and  8  is  32, 
and  7  is  39,  and  6  is  45,  and  5  is  50,  and  4  is  54  ; 
set  down  4,  and  carry  5. 

SecoKd  column,  5  and  4  is  9,  and  8  is  17,  and  4 
is  21,  and  3  is  24,  and  4  is  28,  and  8  is  36,  and  3  is 
39,  and  7  is  46,  and  3  is  49,  and  8  is  57  ;  set  down 
7,  and  carry  5* 

Third  a  !  imn.  5  and  3  is  8,  and  7  is  15,  and  3  is 
18,  and  8  is  26,  and  1  is  27,  and  5  is  32,  and  9  is  41, 
and  2  is  43,  and  b  is  49,  and  2  is  51,  and  7  is  58 ; 

set  down  8,  and  carry  5. 

695874         Fourth  column,  5  and  2  is  7,  and  6  is  13,  and  2 
is  15,  and  5  is  20,  and  2  is  22,  and  6  is  28,  and  5  is 
53,  and  1  is  34,  and  5  is  39,  and  6  is  45  ;  set  down  5,  and 
carry  4. 

Fifth  column.  4  and  9  is  13,  and  5  is  18,  and  7  is  25,  and 
4  is  29,  and  3  is  32,  and  7  is  39,  and  4  is  43,  and  8  is  51,  and 
4  is  55,  and  9  is  64,  and  5  is  69  ;  set  down  69. 

Toia^,  in  figures,  695874;  in  words,  six  hundred  and  nine- 
ty-five thousand,  eight  hundred  and  seventy-four. 

JVo.  4. 

5678         Work  ;  first  column.  4  and  3  is  7,  and  5  is  12, 

9123     and  6  is  18,  and  5  is  23,  and  7  is  SO,  and  6  is  36,  and 

4567     8  is  44,  and  7  is  51,   and  3  is  54,  and  8  is  62;  set 

21 98     down  2,  and  carry  6. 

3456         Second  column.  6  and  4  is  10,  and  6  is  16,  and  7 

1987     is  23,  and  4  is  27,  and  8  is  35,  and  5  is  40,  and  9  is 

2345     49,  and  6  is  55,  and  2  is  57,  and  7  is  64;  set  down 

9876     4,  and  carry  6. 

8765         Third  column.  6  and  2  is  8,  and  7  is  15,  and  8  is 

1203     23,  and  3  is  26,  and  9  is  35,  and  4  is  39,  and  1  is  40, 

2044     and  5  is  45,  and  1  is  46,  and  6  is  52 ;  set  down  2, 

■   '    -  ~     and  carry  5. 

5 1 242         Fourth  column,  5  and  2  is  7,  and  1  is  8,  and  8  is 
16,  and  9  is  25,  and  2  is  27,  and  1  is  28,  and  3  is  31, 

and  2  is  33,  and  4  is  37,  and  9  is  46,  and  5  is  51 ;  set  down 

51. 

Total,  in  figures,  51242 ;  in  words,  fifty-one  thousand,  two 

hundred  and  forty -two. 

B 


1^  Mdiiion. 

JSTo.  5. 

3456         Work  ;  first  column.  9  and  7  is  16,  and  6  is  tft, 

7890  and  8  is  SO,  and  5  is  35,  and  S  is  58,  and  4  is  4ii,  and 

128  6  is  48,  and  9  is  57,  and  6  is  63,  and  8  is  71,  and  9 

907  is  80,  and  7  is  87,  and  7  is  94,  and  8  is  102,  and  6  is 

4017  108  ;  ^et  down  8,  and  carry  10. 
8969         Second  column.  10  and  8  is  18,  and  7  is  25,  and  9 

798  is  34,  and  2  is  S6,  and  5  is  41,  and  8  is  49,  and  7  is 

1476  56,  and  7  is  65,  and  9  is  72,  and  6  is  78,  and  1  is  79, 

5079  and  2  is  81,  and  9  is  90,  and  5  is  95  ;  set  down  5, 

8986  and  carry  9. 

754         Third  column*  9  and  7  is  16,  and  6  is  22,  and  9 

9023  is  3 1,  and  8  is  39,  and  7  is  46,  and  9  is  55,  and  4  i» 

805  59,  and  7  is  66,  and  9  is  75,  and  9  is  84,  and  1  is  85, 

998  and  8  is  93,  and  4  is  97  ;  set  down  7,  and  carry  9, 
1676         Fourth  column.  9  and  6  is  15,  and  2  is  17,  and  1 

2007  is  18,  and  9  is  27,  and  8  is  35,  and  5  is  40,  and  1  is 

6789  41,  and  8  is  49,  and  4  is  53,  and  7  is  60,  and  3  is  63  | 

set  down  63,- 

63758         Total,  in  figures,  63758;  in  words,  sixtj-tlirec 
thousand,  seven  hundred  and  fifty -eight. 

Questions. 

1.  The  number  of  ordained  missionaries  among  the  hea- 
then in  the  year  1821,  was  as  follows  :  From  England,  255  ; 
Scotland,  7;  United  States,  39;  Denmark,  -Z;  Moravians, 
(diff'erent  countries,)  68  :  how  many  in  all  ?  Ans,  351. 

2.  In  the  year  1820,  the  American  Board  of  Commissitmers 
for  Foreign  Missions,  had  the  following  missionaries  and  as- 
sistants among  the  heathen,  to  wit :  In  Eastern  Asia,  25  ; 
Western  Asia,  2  ;  Sandwich  Iblands,  17  ;  American  Indians, 
44  :  how  many  in  all  ?  ^ns.  88. 

3.  The  disbursements  from  the  treasury,  for  expenses,  du- 
ring the  same  year,  were  as  follows :  For  the  Bombay  mission, 
7221  dollars  ;  Ceylon,  7135  ;  Cherokee,  9967  ;  Choctaw, 
10414;  Arkansaw,  1150;  Indian  missions  generally,  252; 
Palestine,  2348  ;  Foreign  niission  school,  3350  ;  Sandwich 
island,  10330  ;  travelling  expenses  of  members  of  the  Board, 
tac.  457  ;  salary  of  Secretary,  500  ;  salary  of  Treasurer,  600  ; 
clerk  hire,  postage  and  stationary,  1143;  printing,  1558; 
agents  to  collect  funds,  261;  expenses  of  meetings,  84; 
transportation  of  articles,  107;  bad  bills,  184;  other  con- 
tingencies, 84 :  how  jnuch  in  all?  ^ns.  57144  dollars. 


Mdition. 


U 


J/ote.  For  further  questions  io  exercise  the  learner*  see  Part  III ;  and  he 
thould  proceed  to  perform  some  of  them  immediately* 

Questions  on  the  foregoing. 

What  are  the  fundamental  rules  of 
\rithmetic  ? 

Whv  are  they  so  called  ? 

What  is  a  Irlition  ? 

Wli'3«i  is  it  culled  simple  addition  ? 

Wha!  is  to  he  observed  in  writing 
down  the  nnnbers  to  be  added  ? 

Which  ■.•olamn  do  vou  add  first  i 

Where  lo  you  bej^in  to  adl  ? 

When  y  .'U  have  adiled  up  the  co- 
lumn of  units,  what  do  you  do  with 
th.;  a^nor.it,  if  it  is  one  fissure  ? 

What,  if  it  is  more  than  one  ? 


When  you  have  added  up  the  column 

of  tens,  and  the  amount  oi  it  is  two 

fissures,    what  is  the  value  of  the 

rij^ht  hand  Ha^ure  ? 
What  of  the  other  ? 
VVhat  do  you  do  with  them  ? 
When  tbe  amount  of  the  column  of 

hundreds  is   two  figures,   what  is 

the  Value  of  each  ? 
VVhat  do  you  do  with  them  ? 
When   you  have  added  up  the  last 

column,  what  do  you  do  with  th« 

amount  ^ 
How  do  you  prove  addition  ? 


^ote.  To  use  the  following  exercise*  let  a  class  be  seated  without  slates  or 
books.  an<l  ai\swer  extetnporaneous'y.  Let  the  mon'tor  take  the  book,  and 
ask,  what  is  the  sum  of  5  and  7  ?  and  look  at  the  cofumn  of  answers,  and  see 
that  the  boy  answers  right.  If  he  answers  wrong,  let  him  put  it  to  the  next, 
but  if  right,  let  him  put  another  question  to  the  next,  as,  what  is  the  sum  of 
8  and  6  ?  carefully  ohserving  not  to  put  the  questions  in  succession  as  they 
stand,  lest  one  answer  sbould  sna'cjest  the  next.  When  a  class  have  been  suf- 
ficiently exercise;l  in  this  way  of  which  the  instructor  will  judge,  let  the  mo- 
nitor vary  tbe  questions,  thus.  7  from  I  %  how  many  remains?  6  from  14,  how 
matiy  remains  ?  -wiA  so  on.  V^'^hen  one  exercise  of  this  kind  has  he/n  attended 
to.  till  most  of  thf  class  can  answer  the  questions  correctly.  let  the  next  ex* 
ercise  he  taken  and  used  in  the  same  manner.  And  if  there  is  reason  to  think, 
at  any  time,  that  the  boys  have  committed  them  to  memoiy,  let  the  monitor 
be  directed  to  make  one  of  his  numbers  larger  or  smaller,  and  observe  that 
the  answer  will  be  as  much  larger  or  smaller;  or,  let  the  instructor  prepare 
new  exercises  of  a  similar  kind,  for  his  monitors  to  make  use  of;  ihat  the 
boys  may  be  compelled  to  [)erform  the  operation  in  their  minds,  before  they 
«an  answer  con-ect'y.  Tt  vdll  greatly  facilitate  the  improvement  ofleamerSf 
for  them  to  have  abundant  exercise  in  this  -toay,  Ferhaps  ffteen  or  tiventy 
mimtieSf  twice  a  day,  ivould  7iot  be  too  much. 


Exerc 

ISE  4. 

2  -f     1  =     3 

2+  11  ==  IS 

3+    9  =  12    4+7  =11 

2+2=4 

2  +  12  =  14 

3  +  10  =  13 

4+8  =12 

2  4-3=5 

3+1=4 

3  +  11  =  14 

4+9  =13 

2  +    4=6 

3+2=5 

3  +  12  =  15 

4  +10  =14 

S+    5  =    7 

3+3=6 

4  +  .  1  =     5 

4  +11  =15 

2+6=8 

3+4=7 

4+2=6 

4  +12.=16 

2  +    7  =    9 

3+5=8 

4+3=7 

5+1  =  6 

2  +     8  =  10 

3+6=9 

4+4=8 

5+2=7 

2  +    9  =  11 

3  +    7  =  10 

4  +    5«=    9 

5+3=8 

^  +  10  «  12 

3+    8  «,  11 

4  +    6  =»  10 

5  +  4  ==.  S 

1« 


Mdition. 


5  and 
5  and 
5  and 
5  and 
5  and 
5  and  10 
5  and  11 
5  and  12 


6  and 
6  and 
6  and 
6  and 
6  and 
6  and 
6  and 
6  and 
6  and 
6  and  10 
6  and  11 

6  and  12 

7  and  1 
7  and 
7  and 
7  and 
7  and 
7  and 
7  and 
7  and 
7  and 
7  and  10 
7  and  11 
7  and  12 


Sand 
8  and 
Sand 
8  and 
8  and 
8  and 
8  and 
8  and 
8  and 


8  and  10 


slO 

s  11 
s  12 
s  13 
s  14 
s  15 
s  16 
s  17 
.s  7 
s  8 
9 
s  10 
s  11 
s  12 
s  13 
s  14 
s  15 
s  16 
s  17 
s  18 
s  8 
s  9 
s  10 
s  IJ 
s  12 
s  13 
s  14 
s  15 
sl6 
.s  17 
sl8 
s  19 
s  9 
s  10 
s  11 
s  12 
s  13 
s  14 
s  15 
s  i6 
.s  17 
s  18 


and  1 1  i 

s  19 

and  12 

IS  20 

and    1 

s  10 

and    2 

IS  11 

and    3  i 

LS  14 

and    4 

IS  13 

and    5 

IS  14 

and    6  ] 

IS  15 

and    7 

IS  16 

and    8 

IS  17 

and    9 

LS  18 

and  10 

IS  19 

and  11 

IS  20 

and  12 

IS  21 

and    2 

IS  13 

and    3 

is  14 

and    4 

IS  15 

and    5 

IS  16 

and    6 

IS  17 

and    7 

is  18 

and    8 

IS  19 

and    9 

is  20 

and  10 

IS  21 

and  12 

IS  23 

and    2 

IS  14 

and    3 

is  15 

and    4 

is  16 

and    5 

IS  17 

and    6 

IS  18 

and    7  i 

IS  19 

and    8 

IS  20 

and    9 

IS  21 

and  10 

IS  22 

and  11 

IS  23 

and  12 

IS  24 

and    2 

is  15 

and    3 

IS  16 

and    4 

is  17 

and    5 

is  18 

and    6 

is  19 

and    7 

IS  20 

and    8 

is  21 

3  and  9 
3  and  10 

3  and  11 
and  12 
and  2 
and 
and 
and 
and 

4  and 
4  and  8 
4  and  9 
4  and  10 
4  and  1 1 

4  and  12 

5  and  2 
5  and 
5  and 
5  and 

and 
and 
and  8 
and  9 
and  10 
and  11 
and  12 


6  and 
6  and 
6  and 
6  and 
6  and 
6  and 
6  and 
6  and 
6  and  10 
6  and  1 1 

6  and  12 

7  and  2 
7  and 
7  and 
7  and 
7  and 


is  22  17+ 

Is  23  17  " 
is  24  17  " 
is  25  17  " 
is  16  17  " 
is  17  17  " 
is  18  18  " 
is  19  18  « 
is  20  18  " 
is  21  18  •* 
is  24ll8  « 
is  23  18  " 
is  24  l8  " 
is  25  18  " 
is  26  18  « 
is  17  18  « 
is  18  18  " 
is  19  l9" 
is  20  l9  " 
is  21  19  " 
is  22  19  " 
is  23  19" 
is  24  19  " 
is  25  19  " 
is  26  19  " 
IS  27  19  " 

18  19" 
is  19  19  " 
is  20  22  " 
is  21  22  " 
'"  22  22  " 

23  22  " 
is  24  22  " 
is  25  23  « 
is  26  23  " 
is  27  23  " 
is  28  23  " 
is  19  23  " 
is  20  24  " 
is  21  24  " 
is  22  24  « 
is  23  24  " 


7=84 

8  "25 

9  "26 

10  "27 

11  "28 

12  "29 

2  "20 

3  "21 

4  "22 

5  "23 

6  "24 

7  "25 

8  "26 
9"  27 

10  "  28 

11  "29 

12  "SO 

2  "21 

3  "22 

4  "23 

5  "24 

6  "25 

7  "26 

8  "27 

9  "28 

10  "29 

11  "  30 

12  "  31 

8  "30 

9  "31 

10  "32 

11  "33 
12"  34 

8  "31 

9  "32 

10  "33 

1 1  "  34 

12  "35 
9  ''  3S 

10  "34 

1 1  "  35 

12  "36 


Addition. 


IT 


^5  and 
M5  and 
25  and 
25  and 
25  and 

25  and 

26  and 
£6  and 
26  and 
26  and 
2u  and 
26  and 

26  and 

27  and 
S7and 
27  and 
27  and 
27  and 
27  and 

27  and 

28  and 
28  and 
S8and 
^8  and 
28  and 
28  and 

28  and 

29  and 
29  and 
29  and 
29  and 


7  is  32  29  and  1 1 


Sis  33 
9  is  34 

10  is  35 

11  is  36 

12  is  37 

6  is  32 

7  is  33 

8  is  34 

9  is  35 
10  is  36 
U  is 


29  and  12 
32  and  7 
32  and  8 
32  and  9 
32  and  1 1 

32  and  12 

33  and  6 
33  and  7 
33  and  8 
33  and  9 
33  and  1 1 

12  is  38  33  and  12 
6  is  33  34  and  7 
8 
9 


7  is  34  34  and 

8  is  35  34  and 

9  is  36  34  and  1 1 

10  is  37  34  and  12 

11  is  38  35  and  7 

12  is  39  35  and  8 

5  is  33  35  and  9 

6  is  34  35  and  11 

7  is  35  35  and  12 

8  is  36  36  and  7 

9  is  37  36  and  8 
1 1  is  39  36  and  9 
12*8  40  36  and  11 

6  is  35  36  and  12 

7  is  36137  and  j 
8is37J37and  8 
9  is  38137  and    9 


s40 
3  41 
39 
40 
4i 
s43 
s44 
s39 
s40 
s  41 
s  42 
.s  44 
s  45 
s  41 
s42 
S  43 
s45 
s46 
s  42 
is  43 
is  44 
is  46 
is  47 
is  43 
is  44 
is  45 
is  47 
is  48 


37  and  11  is  48J47+  8=51 

37  and  12  is  49  47"  9  "'^6 

38  and  7  is  45  47 -11 '*  58 
38  and  8  is  46  47"  12"  59 
38  and  9  is  47  48  "  8  «  56 
38  and  11  is  49  48  "  9  "  5f 

38  and  12  is  50  48  "11  *' 59 

39  and  8  is  47  48  «  12  "6© 
39  and  9  is  48  49  "  8  «  ST 
39  and  11  is  50  49  «  9  "  58 
39  and  12  is  51  49"  11  "60 
42  and  Sis  50  49"  12"  6l 
42  and  9  is  51  53  "  8«6l 
42  and  1 1  is  53  53  «  9  "62 

42  and  12  is  54  53  «  1 1  "  64 

43  and  8  is  51  53  "  12«65 
43  and  9  is  52  54"  7"6l 
43  and  11  is  54  54  «    8  «  62 

43  and  12  is  55  54"    9  "  63 

44  and    Sis  5igl54"ll"65 


44  and  9  is  53 
44  and  1 1  is  55 

44  and  12  is  56 

45  and  8  is  53 
45  and  9  is  54 
45  and  11  is  56 

45  and  12  is  57 

46  and  8  is  54 
is  44]46  and  9  is  55 
is  45J46  and  1 1  is  57 
is  46  46  and  12  is  58 


54  "  12"  66 

55  "  7  "  62 
55  "  8  "  63 
55  "  9  "  64 
55  "  1 1  "  66 
55  "  12  "  67 
64  "  9  "  73 
75  "  1 1  "  86 
86  "  12  "98 
74"  9  "83 
87"    6  "93 


Tell  the  sum  of 
S+5+2     Ms.  10 
10     6     3  19 

4  7     2  13 

5  8     3  16 
7     4     9             20 


Exercise  5^ 


S     9     4. 
4    .8     2 


21 


6+7+3     Ms.  16    3  +  10+12 
8     6     5  19    3       9       7 

7     5     4  161  4     11       6 


9  4  6 

5  3  7 

3  2  8 

14  3  5  8 


19 
15 
13 

m  7 


1® 
21 
J  6 

17 


12 


:S 


iiddttioii. 


Tfell  the  suth 

HiXERCISE   C 

of 

) 

9+  ^+-7  ^ws.  18 

5+  7+  9 

dns 

.  21 

5+   7+12 

Ans,  S4 

8       3     1^ 

23 

2     1^       9 

23 

6       7      11 

24 

2     11       3 

16 

2       6      11 

19 

6       8      12 

26 

7       4       6 

17 

3      114 

18 

7       5       8 

20 

3       9       5 

17 

4       7      11 

22 

7       6     12 

^5 

6       7       4 

17 

4       9      12 

25 

8     115 

24 

4       6     11 

21 

5       6      11 

22 

8     12       3 

23 

SUBTRACTION, 

Is  faking  a  less  number  from  a  greater,  so  as  to  find  their 
difference,  which  is  called  the  remainder. 

It  is  called  Simple  Subtraction,  when  the  numbers  are  of 
one  denomination. 

HuLE. 

1.  Write  the  less  number  ur^er  the  greater,  in  such  a 
manner  that  units  shall  stand  under  units,  tens  under  tens, 
&c.  and  draw  a  line  under. 

2.  Begin  at  the  right  hand,  and  take  the  lower  figure  from 
that  above  it,  and  set  down  the  remainder  underneath  ;  and 
so  on,  with  all  the  rest. 

3.  But  if  the  lower  figure  is  greater  than  that  ^bove  itj 
borrow  10  and  add  to  the  upper  figure,  and  then  subtract  the 
lower,  and  set  down  the  remainder. 

4.  And  where  you  borrow  10  to  add  to  ilie  upper  fi2;ure5 
in  bne  column,  carry  or  add  1  to  the  lower  figure  of  the  next 
column. 

JK'ote.  The  reason  trhy  1  is  carried  while  10  is  borrowed,  is,  that  I  in  tb^ 
ftolumn  of  tens  is  eq«al  to  10  in  the  column  of  units.  And  if  10  is  added  to 
the  upper  line  in  the  column  of  units,  and  I  is  added  to  the  lower  line  in  the 
colunin  of  tens,  an  equal  amount  is  added  to  each  line.  But  if  the  same 
amount  is  added  to  each  line,  their  difference  will  remain  the  same.  Thus-, 
the  difference  between  8  and  6  is  2 ;  and  if  you  add  10  to  eacli,  they  v/iil  b« 
18  and  16,  and  their  difference  will  still  be  2. 
Example. 

Find  the  difference  between  3456  and  1238. 
3456         Here,  I  first  set  down  3456,  the  greater  number, 
1238     and  then  under  it,  J 238,  the  less  ;  so  that  units  stand 

under  units,  &c.    Then  I  begin  at  the  right  hand,  and 

^218     say,  8  from  6  T  cannot ;  borrow  10,  and  add  to  the  6, 

and  it  makes  1 6 ;  8  from  1 6  leaves  8 ;  set  down  8. 

Having  added  10  to  tlie  column  of  units  in  the  upper  line>  I 

must  add  as  much  to  the  lower,  and  so  I  carry  1  to  the  columia 


Subtraction.  If 

©f  tens,  and  say,  1  to  3  is  4,  and  4  from  5  lea.ves  1  ;  set  down 
1 ,  and  proceed  to  the  next  column.  2  from  4  leaves  2 ;  set 
down  2,  and  proceed  to  the  next.  1  from^  ajeaves  2 ;  set 
down  2.     And  the  remainder,  thus  found,  is  2218. 


Add  the  remainder  to  the  less  number,  and  if^  the  work  i* 
right,  their  sum  will  equal  the  greater  number. 

J\ote,  The  following  examples  are  to  be  perfoi^med  by  a  class  with  a  mo- 
nitor, as  those  in  addition. 

JSTo.  1. 

From  2345         Work,  Begin  at  the  right  hand  at  the  bot- 

Take   1452     torn,  and  say,  2  from  5  leaves  3  ;  set  down  3. 

Second  column.  5  from  4  I  cannot ;  borrow  1 0, 

Rem.  893  which  added  to  the  4  is  14  ;  5  from  14  leaved 
9  ;  set  down  9.  Third  column.  For  the  10  that 
I  borrowed  in  the  last  column,  carry  I  to  this  ;  I  and  4  is  5  ; 
5  from  3  I  cannot ;  borrow  10,  and  say,  5  from  13  leaves  8 ; 
set  down  8.  Fourth  column.  Carry  1  to  1  is  2 ;  2  from  2 
leaves  0. 

Remainder,  in  figures,  893 ;  in  words,  eight  hundred  and 
ninety-three. 

JS^o.  2. 

From  98764         Work.  5  from  4  I  cannot;  borrow  10,  and 

Take  34985     5  from  14  leaves  9^  set  down  9.     1  carried  to 

8  is  9  ;  9  from  6  I  cannot ;  borrow  10,  and  9 

Rem.  63779     from  16  leaves  7;  set  down  7.     1  carried  to  9 
is  10  ;  10  from  7  I  cannot ;  borrow  10,  and  10 
from  17  leaves  7 ;  set  down  7.     1  carried  to  4  is  5  ;  5  from 
8  leaves  3  ;  set  down  3.  3  from  9  leaves  6 ;  set  down  6. 

Remainder,  in  figures,  63779  ;  in  words,  sixty- three  thou- 
sand, seven  hundred  and  seventy- nine. 

^0.  3. 

From  91234         Work.  1  from  4  leaves  3  ;  set  down  3.    0 

Take  51301     from  3  leaves  3  ;  set  down  3.     3  from  2  I  can- 

not ;  borrow  10,  and  3  from  12  leaves  9  ;  set 

Rem,  39933     down  9.     1  carried  to  lis  2,  2  from  1  I  can- 
not; borrow  10,  and  2  from   11  leaves  9 ;  set 
down  9.     1  carried  to  5  is  6 ;  6  from  9  leaves  3  ;  set  down  3, 
Remainder,  in  figures,  39933  ;  in  words,  thirty -nine  thou- 
BStnd,  nine  hundred  and  tiiirty-three^ 


2» 


Subtraction^ 


JVo.  4. 

From  98r6545         Work,  4  from  3  I  cannot,  but  4  from  IS 

Take     987654     leaves  9.     1  to  5  is  6;  6  from  4  I  cannot, 

. but  6  from  14  leaves  8.     1  to  6  is  7 ;  7  from 

Rem.  8888889  5  I  cannot,  bat  7  from  15  leaves  8.  1  to  7 
is  8 ;  8  from  6  I  cannot,  but  8  from  l6  leaves 
8.  1  to  8  is  9  ;  9  from  7  I  cannot,  but  9  from  17  leaves  8. 
1  to  9  is  lU ;  10  from  8  I  cannot,  but  10  from  18  leaves  8.  I 
to  0  is  1  ;  I  from  9  leaves  8.  Remainder,  8888889. 
Questions. 

1.  If  the  population  of  the  world  is  820000000,  and  the 
number  of  nominal  christians  is  214000000,  how  manj  are 
destitute  of  the  gospel  ?  Jlns.  606000000. 

2.  Our  Lord  and  Saviour,  previous  to  his  ascension,  in  the 
year  33,  commanded  his  disciples  to  preach  the  gospel  among 
all  nations,  and  the  London  Missionary  Society  was  formed 
in  the  year  1795  ;  how  long  between  ?        Ans.  1762  years. 

3.  The  canon  of  scripture  was  completed  in  the  year  97, 
and  the  British  and  Foreign  Bible  Society  was  formed  in  the 
year  1804  ;  how  long  between  ?  Ans,  1707  years. 

Questions  on  the  foregoing. 


What  is  Subtraction  ? 

When  is  it  called  SimpleSubtraction? 

How  many  numbers  are  employed  ? 

How  must  they  be  written  down  ? 

Where  do  you  begfin  ? 

What  is  to  be  done   if  the  upper 


figure  is  less  tlian  the  lower  ? 
When  you  have  borrowed  10.  how 

many  must  you  carry,  and  where  ? 
Why  is  1  carried,  while  10  was  bcw- 

rowed  ? 
How  18  subtraction  proved  ? 

Exercise  7. 


From    Take 

Ans. 

Prom 

Take 

Ans. 

From 

Take 

Ans. 

12,        3+   4, 

5 

n, 

4+3+2, 

2 

20, 

6+5  +  3, 

6 

15,        4        7, 

4 

ir. 

5     3     4, 

5 

18, 

9     3      4, 

2 

18,       9       6, 

S 

19, 

7     2     3, 

7 

19, 

4     5     7, 

S 

20,       7     12, 

1 

21, 

9     6     S, 

2 

ir. 

5      1     4, 

r 

25,       9     12, 

4 

16, 

4     3     7, 

£ 

16, 

4     3     6, 

3 

U,       3       5, 

6 

23, 

5     3     4, 

11 

15, 

1     2     3, 

9 

12,   1+2+6, 

3^ 

24, 

6     9     3, 

6 

19, 

2     4     7, 

6 

MULTIPLICATION, 

Is  a  short  way  of  performing  addition,  and  teaches  how  to 
find  the  amount  of  a  number  when  added  to  itself  a  certain 
number  of  times. 

It  is  called  Simple  Multiplication,  when  the  number  to  be 
iaaultiplied  is  of  one  denomination, 


Multiplication. 


21 


m 


The  number  to  be  multiplied  is  called  the  multiplicand  s 
the  number  to  multiply  by,  is  called  the  multiplier  ;  and  the 
number  found,  or  total  amount,  is  called  the  product. 
The  multiplicand  and  multiplier  are  both  called  factors. 
J>tote.  Before  pre)ceeding  to  any  operatious  in  multiplication,  it  is  necessary 
to  learn  perfectly  the  following  table. 

Multiplication  Table. 


%  times 

1 
4  times 

6  times 

2  is    4 

2  is     8 

2  is  12 

Sis    6 

3  is  12 

Sis  18 

4  is    8 

4  is  16 

4  is  24 

.5  is  10 

5  is  20 

5  is  30 

6  is  \2 

6  is  24 

6  is  36 

ris  14 

7  is  28 

7  is  42 

Sis  16 

8  is  32 

8  is  48 

9  is  18 

9  is  36 

9  is  54 

10  is  20 

10  is  40 

10  is  60 

1 1  is  22 

i  1  is  44 

11  is  66 

12  is  24 

12  is  48 

12  is  72 

3  times 

5  times 

7  times 

Sis    6 

2  is  1 0 

2  is  14 

3  is    9 

3  is  15 

3  is  21 

4  is  1£ 

4  is^O 

4  is  28 

5  is  15 

5  is  25 

5  is  35 

6  is  18 

6  is  30 

6  is  42 

7  is  21 

7  is  35 

7  is  49 

8  is  £4 

Sis  40 

8  is  56 

9  is  27 

9  is  45 

9  is  63 

10  is  30 

10  is  50 

10  is  70 

1 1  is  33 

1 1  is  55 

11  is  77 

12  is  36 

12  is  60 

12  is  84 

8  times 
2  is    16 


3  IS 

4  is 

5  is 

6  is 

7  is 

8  is 

9  is 
10  is 
U  is 
12  is 


24 
32 
40 
48 
56 
64 
72 
80 
88 
96 


9  times 
2  is    18 


3  is 

4  is 

5  is 

6  is 

7  is 

8  is 

9  is 

10  is 

11  is 


27 
36 
45 
54 
63 
72 
81 
90 
99 


12  is  108 


10  times 

2  is    20 

3  is 

4  is 

5  is 

6  is 

7  is 

8  is 

9  is 

10  is  100 

11  is  110 

12  is  120 
1 1  times 

2  is    22 

3  is 

4  is 

5  is 

6  is 

7  is 

8  ih 

9  is 

10  is  no 

11  is  121 

12  is  132 


33 
44 
SB 

66 
77 
88 
99 


12  times 


2 

is 

24 

3 

is 

36 

4 

IS 

48 

5 

is 

60 

6 

is 

72 

7 

is 

84 

8 

is 

96 

9 

is 

108 

10 

is 

120 

11 

is 

132 

12 

ia 

144 

Case  I. 
To  multiply  any  given  number  by  a  single  figure,  or  by  any 
number  not  over  12. 

Rule. 

1.  Set  down  the  multiplier  under  the  units  figure,  or  right 
hand  place  of  the  multiplicand,  and  draw  a  line  underneath. 

2.  Begin  at  the  right  hand  figure,  and  Hiultiply.  If  the 
product  of  the  units  figm^e  of  the  multiplicand  is  but  one 
figure,  set  it  down  in  the  place  of  units.  If  it  is  more  fig- 
ures than  one,  set  down  the  right  hand  figure,  or  units,  and 
«arry  the  rest  to  be  added  to  the  product  of  the  ten». 


2t  JIultiplication* 

S.  Multiplj  the  tens,  and  to  the  product  add  what  wai 
earned  from  the  product  of  the  units.  Set  down  the  right 
Sland  figure,  and  carry  the  rest  to  be  added  to  the  product  of 
the  hundreds.  And  so  on,  to  the  end,  setting  down  the  whole 
m  the  last  place. 

JVbie.  To  prevent  mistakes,  it  will  be  well  to  make  t  miaute  of  what  is 
to  be  earned  each  time. 

Example  \, 
Multiplj  5678  TTork.     4  times  8  is  52  ;  set  down  2,  and 

Bj        4     carry  3,     4  times  7  is  28,  and  3  I  carried  is 

Prod.  22712  31  ;  set  down  I,  and  carry  3.  4  times  6  is 
24,  and  5  I  carried  is  27;  set  down  7  and 
carry  2.  4  times  5  is  20,  and  2  I  carried  is  22  ;  set  down 
22.  Product,  in  figures,  22712;  in  words,  twenty-two 
thousand  seven  hundred  and  twelve. 

JVbtf*.  Observe,  th;jt  I  do  not  carry  the  tens  of  the  product  of  the  units  t« 
the  tens  of  the  multiplicand,  before  I  multiply  them,  but  to  their  product  j 
Ihe  reason  of  which  will  appear,  by  varying  the  process,  as  follows  : 

5678 
4 

'  Here,  4  times  8  is  32,  4  times  70  is  280,  4  times 

32  600  is  2400,  and  4  times  5000  is  20000,  which  added 

280  together,  is  2271 2,  as  before.     From  which  itappears, 

2400  that  the  carrying  is  only  done  in  the  addition  of  the 

20000  ser^ral  products  together. 

22712 

Example  2. 
Multiply  34567         Work.    7  times  7  is  49:    set  down  9, 
By  7     and  carry  4.     7  times  6  is  42,  and  4  I  car- 

ried,  is  46  ;    set  down  6,  and  carry  4.     7 

Prod.       241969     times  5  is  35,  and  4  I  carried,  is  39  ;    set 
down  9,  and  carry  3.     7  times  4  is  28, 
and  3  I  carried,  is  31  ;  set  down  1,  and  carry  3.     7  times  3 
ia  21,  and  3  I  carried,  is  24  ;  set  down  24.     Prod.  241969. 

Example  S. 
Multiply  9876         Work.     12  times  6  is  72  ;  set  down  2, 
By  12     and  carry  7.     12  times  7  is  84,  and  7  I  car- 

ried  is   91  ;    set   down    1,   and    carry  9. 

Prod.     118512     12  times  8  is  96,  and  9  I  carried,  is  105  ; 

set  down  5,  and  carry  10.     12  times  9  is 

108,  and  10  I  carried  is  118 ;  set  down  118.    Prod.  1185  U* 


Multiplication.  tt 

Case  II* 
To  multiply  by  a  number  consisting  of  eevcral  figures. 

Rule* 

1.  Set  down  the  multiplier  under  the  multiplicand,  so  that 
units  shall  stand  under  units,  tens  under  tens,  &c.  and  draw 
a  line  underneath. 

2.  Multiply  the  whole  of  the  multiplicand  bf  the  first  or 
units  figure  oi  the  multiplier,  and  set  down  the  product,  a» 
in  case  first. 

S.  Multiply  the  whole  of  the  multiplicand  by  the  second 
figure  of  the  multiplier,  and  set  down  the  product  in  the 
game  manner,  only  placing  each  figure  of  the  product  one 
remove  to  the  left. 

4.  Multiply  in  the  same  manner  by  the  third  figure  of  the 
multiplier,  and  place  the  figures  of  the  product  two  removes 
to  the  left.  And  so  on,  to  the  end,  placing  the  figuies  of 
each  product  so  that  the  first  shall  always  stand  under  the 
figure  by  which  you  are  multiplying. 

5.  W  hen  you  have  nmltipliett  by  all  the  figures  of  the 
multiplier,  add  the  several  products  together,  and  their  sum 
will  be  the  answer,  or  whole  product  required. 

Examyle  1. 
Multiply     3456         Work.     First  fgure,     5  times  6  is  20 5 
By  3-i5     set  down  0,  and  cany  S.     5  times  5  is  2  ', 

and  3  1  canitd,  is  28  ;    set  down  8,  nnd 

17280     carry  52.     5  times  4  is  20,  and  2  1  carried 

6912       is  22  ;    set  down  2,  and  carry  2.     5  times 

10368         Sis  15,  and  i2  J  carried,  is  1 .  ,  srt  down  17* 

Second  fgure,     2  times  6  is  1^;    set 

Prod.     112S20O     down  9  under  the  2,  and  carry  i.   2  times 
5  is   10,  and  1  1  carried  is  1  I  ;    ^et  down 
1,  ami  carry  1.     2  times  4  is  8,  and   1  1  earned  is  9;    set 
down  9.     2  times  3  is  6  ;  set  down  6. 

Third  figure.  3  times  6  is  18;  set  down  8  under  the 
5,  and  carry  1 .  3  times  5  is  15,  and  1  I  carried  is  1 6  ;  set 
down  6,  and  carry  1.  3  times  4  is  1£,  and  1  1  carried,  is  13  ; 
get  down  3,  and  carry  1.  3  times  3  is  9,  and  I  1  carried  is 
10  ;  set  down  10. 

Having  obtained  the  product  of  the  multiplicand  by  each 
figure  of  the  multiplier,  those  products  are  now  to  be  iidded. 
Draw  a  line  under  and  add. 


24  Multiplication^ 

First  eolufnn,  0  is  0  ;  set  down  0.  Second  column,  2 
and  8  is  10:  set  down  0,  and  carry  1.  Third  column.  1 
carried  to  8  is  9,  and  I  is  10,  and  2  is  12 ;  set  down  2,  and 
carry  1.  Fourth  column.  1  carried  to  6  is  7,  and  9  is  16, 
and  7  is  23  ;  set  down  S,  and  carry  2.  Fijih  column.  2 
carried  to  3  is  5,  and  6  is  11,  and  1  is  12  ;  set  down  2,  and 
carry  1.  Sljjth  column.  1  carried  to  0  is  1  ;  set  down  1. 
Seventh  column.    1  is  1  ;  set  down  I.    Total  Prod.  1 12S200. 

J\  oife.  1  he  reaon  for  placing  the  fiict  lUrure  of  ihc  second  product  one  re- 
move to  the  left,  or  in  the  place  of  tens,  is,  that  it  is  ilie  product  of  the  units 
of  the  multiplicand  by  the  tens  of  the  muitiplien  and  is  the  efore  tens,  and 
should  hf^  put  in  the  place  of  tern.  Fo  the  same  reason,  the  first  figure  of* 
cf  the  tliird  product  is  hundreds,  and  of  the  fourth  4  housa  ds»  and  soon. 
Accordingly*  in  the  abov  example,  17280  is  the  product  of  3456  by  5.  But 
6912»  with  the  2  standir  *<  iu  \hf  place  of  tens,  is  th^-  same  as  691  £0  and  is  the 
product  of  3456  by  20.      '  s   uith  th;-  K  stand  in  ibe  place  of  hundreds, 

is  the  same  as  1036801  produci.  of  3456  by  300. 

Example  ^.  ' 

Multiply     98765  Work.     First  figure.  8  times  5  is  40  } 

By  678     set  down  0  under  the  8,  and  carry  4. 

8  times  6  is  48,  and  4  I  carried,  is  52  ; 

790120     set  down  2  and  carry  5.     8  times   7  is 

691355       56,  and  5  I  carried,  is  61  ;    set  down  1, 

592590         and  carry  6.     8  times  8  is  64,  and  6  I 

carried,  is  70  ;    set  down  0  and  carry  7. 

Prod.  66962670  8  times  9  is  7£,  and  7  I  carried,  is  79; 
set  down  79. 
Second  figure.  7  times  5  is  35 ;  set  down  5,  under  the  7, 
and  carry  3.  7  times  6  is  42,  and  3  1  carried,  is  45 ;  set 
down  5,  and  carry  4.  7  times  7  is  49,  and  4  1  carried  is  53 ; 
«et  down  3,  and  carry  5.  7  times  8  is  56,  and  5  1  carried,  is 
61 ;  set  down  1,  and  cany  6.  7  times  9  is  63,  and  6  1  car- 
ried, is  69  ;  set  down  69. 

Third  figure.  6  times  5  is  30  ;  set  down  0,  under  the  6, 
and  carry  3.  6  times  6  is  36,  and  3  1  carried,  is  S9 ;  set  down 
9,  and  carry  3.  f  times  7  is  42,  and  3  1  carried,  is  45  ;  set 
down  5,  and  carr/  4.  6  times  8  is  48,  and  4  I  carried,  is 
52  ;  set  down  2,  and  carry  5.  6  times  9  is  54,  and  5  1  car- 
ried, is  59 ;  set  down  59. 

Next,  draw  a  line  under  and  add.  0  is  0  ;  set  down  0. 
5  and  2  is  7  ;  set  down  7.     5  and  1  is  6  ;  set  down  6.     9  and 

5  is  12;  set  down  'z,  and  carry  i.  1  to  5  is  6,  and  I  is  7, 
and  9  is  16  ;  set  down  6,  and  carry  1.  1  to  2  is  3,  and  9  is 
12,  and  7  is  19  ;  set  down  9,  and  carry  1.     1  to  9  is  iO,  and 

6  is  16 ;  set  down  6,  and  carry  1.     1  to  5  is  6 ;    set  down  6. 

Total  product,  66962670^ 


Multiplicati9n,  25 

Case.  III. 
When  ther«  arc  cyphers  at  the  right  hand  of  one  or  botk 
the  factors,  or  between  the  other  figures  of  the  mtiltiplier. 

Rule. 

1 .  If  the  cyphers  are  at  the  right  hand  of  the  numbers, 
multiply  the  other  figures  as  in  case  I  or  2,  and  annex  to  the 
product  as  many  cyphers  as  were  neglected. 

2.  If  cyphers  are  between  other  figures  in  tlie  multiplier, 
neglect  them  also,  only  take  care  to  place  the  first  figure  of 
every  product  exactly  under  its  multiplier. 

Example. 
Multiply       304050  Work,    Neglect  the  three  cyphers 

By  20 SCO       at  the  right  hand  of  the  factors,  and 

begin  with   3  and  5.     S  times  5   is 

91215000     IS  ;    set  down   5,  under  the  3,  an- 
60810      '         nexing  three  cyphers  to  it,  and  car- 

— — . ry  1.     3  times  0  is(),  and   I  I  carri- 

Prod.  6172^^15000  ed,  is  1  ;  set  down  i.  3  times  4  is 
\2;  set  down  2,  and  carry  1.  3 
times  0  is  0,  and  I  I  carried,  is  I  ;  set  down  i.  3  times  3  is 
9.  The  next  figure  of  the  multiplier  being  a  cypher,  neglect 
that,  and  take  2.  2  times  5  is  10;  set  down  0,  under  2,  and 
carry  1.  2  times  0  is  0,  and  1  I  carred  is  i  ;  set  dtwn  I. 
2  times  4  is  8  ;  set  down  8.  2  times  0  is  0  ;  set  down  0. 
2  times  3  is  6  ;  set  down  6. 

Next,  draw  a  line  under,  and  add.     0   is  0  ;  set  down  0. 

0  is  0  ;  set  down  0.  0  is  0  ;  set  down  0.  5  is  5  ;  set  down 
5.     1  is  1  ;  set  down   1.     0  and  2  is  2 ;  set  down  2.     1  and 

1  is  S  ;  set  down  2.  8  and  9  is  17  ;  set  dov/n  7,  and  carry 
1.  1  to  0  is  1  ;  set  down  1.  6  is  6  ;  set  down  6.  Total 
^ro^we^,  6172215000. 

J\ote.  The  reas^.  for  annexing  to  the  product  Dieycphfrs  which  had  been 
neglected  at  the  rii^ht  hand  of  the  factors,  is,  tliat  in  the  above  exaruplf ,  the- 
product  of  the  first  sigrilficant  figures  is  not  reaif.y  the  ])roduct  of  5  Wy  3,  bus, 
of  50  bv  300.  wljicli  is  not  15,  I.mi  1.  HK).  A;u]  the  rerso  :-  r  i-c'-i^ihe first 
figure  of  the  xjroduct  of  5  by  il  it;  Hic  place  of  hundreos  c-  tho  sands,  is,  that' 
it  is  not  really  the  product  of  5  by  ±,  hvX  of  50  by  20000,  w  hich  is  not  10,  but? 
1000000. 

Proof. 
First  meikocL  Multiply  the  multiplier  by  the  multiplicand, 
and  the  product  will  be  the  same  as  before,   if  the  work  is 


right. 


C 


2^  Multiplicatiou. 

Exnmple,  Take  the  first  example  under  Case  I.  and  invert 
the  factors. 

Multiply  4  Work.  8  times  4  is  32;  set  down  31.  7 
By  5678  times  4  is  .28  ;  set  down  28,  so  that  the  8  shall 
stand  under  the  7.  6  times  4  is  24  ;  set  down 
24,  so  that  the  4  shall  stand  under  the  6.  5 
times  4  is  20 ;  set  down  20,  so  that  the  0  shall 
stand  under  the  5.  Next,  add,  2  is  2 ;  set 
down  2.  8  and  3  is  1 1  ;  set  down  1,  and 
carry  1 .  1  to  4  is  5,  and  2  is  7 ;  set  down 
Prod.  22712  7.  0  and  2  is  2  ;  set  down  2.  2  is  2  ;  set 
down  2.  l^otal  product,  22712,  which  is  the 
same  as  before ;  so  I  conclude  the  work  is  right. 

Second  method.  Divide  the  product  by  the  multiplier,  and 
the  quotient  will  be  equal  to  the  multiplicand,  if  the  work  is 
right.  This  is  the  method  usually  practised  by  experienced 
arithmeticians,  as  the  safest;  but  as  the  learner  is  not  sup- 
posed to  be  as  yet  acquainted  with  division,  let  him  prove  his 
work  by  the  ftrst  method,  or  let  him  make  use  of  the 

Third  method,  1.  Cast  the  9's  out  of  both  the  factors,  and 
place  the  remainders  at  the  opposite  ends  of  a  dotted  line.  2. 
Multiply  those  remainders  together,  cast  the  9's  out  of  their 
product,  and  set  down  the  remainder  above  the  dotted  line. 
3.  Cast  the  9's  out  of  the  product  you  wish  to  prove,  and  set 
down  the  remainder  under  the  dotted  line.  If  the  work  is 
right,  the  figures  above  and  below  the  dotted  line  will  be  alike. 

Example,.  Proof, 

Multiply  34562  1 

By  5  Md,  2 5  Mr. 

1 

Product,  172810  Fred. 

Explanatiov.,  I  first  take  the  multiplicand,  and  say,  3  and 

4  is  7,  and  5  is  12;  cast  away  9,  and  3  remains.  3  and  6  is 
9  ;  cast  it  away.  2  is  left,  as  the  remainder  of  the  multipli- 
cand, which  I  set  down  at  the  left  hand  of  the  dotted  line.  I 
then  take  the  multiplier,  which  being  but  5,  there  is  no  9  to 
cast  away  :  so  1  set  down  5  at  the  right  hand  of  the  dotted 
line,  as  the  remainder  of  the  multiplier. 

2.  I  multiply  these  twn  remainders  together,  saying  2  times 

5  is  10 ;  cast  away  9,  and  1  remains,  which  I  set  down  above 
the  dotted  line. 


Multiplioatiou, 


m 


%  I  take  the  product,  and  say,  1  and  7  is  8,  and  2  is  10; 
«ast  away  9,  and  1  remains.  1  and  8  is  9  ;  cast  it  away.  1 
and  0  is  I ,  whicli  being  less  than  9,  I  set  it  down  under  the 
dotted  line,  as  the  remainder  of  the  product. 

And  the  figures  above  and  below  the  dotted  liae  being 

alike,  I  conclude  the  work  is  right. 

jstote.  This  method  of  proof  is  not  infallible,  because  the  right  figures  may 
stand  in  the  product,  and  not  stand  in  the  right  order  ;  or  two  wrong  figiu-et 
may  amount  to  the  same,  when  added  together,  as  the  two  right  ones  would. 
But  as  this  method  will  usually  detect  a  mistake,  and  is  shorter  than  the  other 
methods,  it  is  thought  useful  to  he  retained. 

Questions. 

1.  If  a  child  eats  6  cents  worth  of  fruit,  nuts,  &c.  every 
week,  how  many  cents  could  he  save  in  a  year,  being  52 
weeks,  and  give  for  the  education  of  heathen  children,  by  de- 
nying himself  those  indulgencies  ?  •^HS.  312. 

2.  If  a  child  should  employ  his  play  hours  in  working^for 
the  education  of  heathen  children,  and  should  earn  S  cents  a 
day,  by  so  doing,  how  many  cents  would  he  earn  in  a  year 
for  that  purpose,  there  being  S13  working  days  ?  Ans,  939. 

3.  If  the  population  of  the  United  States  is  9630000,  and 
every  person  should  earn  or  save  3  cents  a  day  for  doing  good, 
liow  many  cents  would  that  be  in  a  year  ?    Jin^,  9042570000* 

Questions  on  the  foregoing. 

rather  than  under  the  figure  muU 
tiplied  ? 

When  you  have  multiplied  by  all  th« 
figures  of  the  multiplier,  what  do 
you  do  next  ? 

What  is  the  third  case  ? 

What  do  you  do  with  cyphers  at  th« 
right  hand  of  the  factors  ? 

Why  do  you  annex  th^m  to  the  right 
hand  of  your  product  ? 

What  do  you  do  with  c\  phers  be- 
tween other  figures  ©f  your  multi- 
plier ? 

What  is  the  first  method  of  proof  I 
second  ? 

Which  is  the  safest  ? 

What  is  the?  third  method  of  proof  ? 

In  the  third  method,  what  is  the  first 
thing:  to  be  done  ?  the  second  }  the 
third  ? 

When  do  you  conclude  the  work  '\% 
right  ? 

Is  this  pi^thod  of  proof  certain  } 

Why  so  \ 

Wh^  then  is  it  retaiaed  ? 


What  is  multiplication  } 

When  is  it  called  simple  multiplica- 
tion ? 

How  many  numbers  are  employed  ? 

What  is  each  one  called  ? 

What  are  they  called  together  ? 

What  is  the  answer  called  ? 

What  is  to  be  done,  before  you  be- 
gin to  perform  questions  in  multi- 
plication ? 

What  is  the  first  case  } 

How  must  the  numbers  be  written 
down } 

Where  do  you  begin, &  how  proceed? 

When  you  carry,  tp  what  do  you  add 
the  figure  carried  ? 

Why  so  ? 

What  is  the  second  case  ? 

How  many  figures  of  the  multiplier 
do  you  use  at  a  time  ? 

When  you  have  multiplied  the  first 
figure  of  the  multiplicand  by  the 
second  figure  of  the  multiplier, 
where  do  you  begin  to  set  it  down  P 

Why  do  you  get  it  under  that  figure, 


2$ 

Multiplication, 

Exercise  8. 

Tell  the  amount  of 

5x2x2,       Ms,   123X3X5, 

Ans.  45 

5x3x2,       Ms 

.  SO 

3    3    2, 

18 

3    4    5, 

60 

5    2    4, 

40 

2    2    4, 

16 

3     2     6, 

36 

5     5    4, 

100 

2    3    3, 

18 

3    2    4, 

24 

6    2    2, 

24 

2    3    4, 

24 

3     3     3, 

27 

6    3    2, 

36 

2    2    5, 

20  4    3     3, 

36 

6    4    2, 

48 

2    3    5, 

30  5    2    2, 

20 

6    4    3, 

72 

Exercise  9, 

Tell  the  amount  of 

2X0X10,  Ms. 

10 

4x2x4, 

Ms.  32 

8x  5x1       Ms 

.  40 

2    3      8, 

48 

5     2     5, 

50 

9      4    2, 

72 

3    4      8, 

96 

5     2     2, 

20 

9     10    2, 

180 

5-6      2, 

6Q 

6     5     2, 

60 

2       1     9, 

18 

5    5      2, 

50 

7    2     1, 

14 

3       3    9, 

81 

S    5      5, 

75 

8    3     2, 

48 

2      5    9, 

90 

3    4      4, 

48|8    4    3, 

96 

4    10    3, 

120 

Exerc 

JISE   10. 

From    take 

./Itw.     From    take           ^ns*     From    take 

^nsi 

24,     4x  5, 

4 

36,     9X 

3,           9 

37,     7X  4, 

9 

28,     3       5, 

13 

45,     3 

12,           9 

41,     6      5, 

11 

20,     2      7, 

6 

24,     4 

3,          12 

29,     7       3, 

8 

19,     3      4, 

7 

28,     4 

4,         12 

36,     8       4, 

4 

14,     5       g, 

4 

32,     5 

3,         17 

19,     3       4, 

7 

>1,     7      4, 

3 

18,     3 

5,            3 

25,     8       2, 

9 

..r,    2     8, 

11 

22,     5 

3,           7 

38,     2     11, 

16 

Exercise  11. 

From        take 

»^7is.       From     ta 

ke          ^7is.      From     take 

.fns. 

iOO,     8x   8, 

36 

7^,     CX12,        24 

83,     6x12, 

13 

96,     4    12, 

48 

67,     8 

b,       19 

94,     8      9, 

22 

72,     7      9, 

9 

98,     6 

11,         Sz 

105,     6    11, 

39 

66,     9       5, 

21 

102,     9 

8,       SO 

86,     9      5, 

41 

48,     6      5, 

18 

55f     6 

r,     13 

74,     7      6, 

32 

84,     5     12, 

24 

48,     4 

7,       20 

'  65,     5       7, 

30 

#r.   7    7, 

18 

79,     7 

8,       23 

47,     7       5, 

12 

Division.  3§ 

DIVISION, 

Is  a  short  method  of  performing  subtraction,  and  teaches 
to  find  how  often  one  number  is  contained  in  another. 

It  is  called  Simple  Division,  when  the  number  to  be  di- 
vided is  of  one  denomination. 

Tlie  number  to  be  divided,  is  called  the  dividend ;  the 
number  to  divide  by,  is  called  the  divisor  ;  and  the  result,  or 
answer,  is  called  the  quotient, 

j\ote  Before  proceeding  to  perform  operations  in  division,  let  the  student 
learn  the  multiplication  tab  e  in  an  inverted  order,  as  t'ollows :  2  in  4  i« 
twice,  2  in  6  is  3  times,  2  in  8  is  4  times,  kc. 

Rule. 

1.  Set  down  the  dividend  first,  and  the  divisor  at  the  left 
hand  of  it,  separated  by  a  curve  line ;  and  leave  a  place  at 
the  right  to  set  the  quotient,  separated  also  from  the  dividend 
by  a  curve  line. 

2.  Take  the  fewest  figures  of  the  dividend,  beginning  at  the 
left  hand,  that  will  contain  the  divisor,  and  see  how  many 
times  they  will  contain  it,  and  place  the  figure  denoting  ihat 
Dumber  of  times,  for  the  first  figure  of  the  quotient. 

3.  Multiply  the  divisor  by  that  quotient  figure,  and  place  ' 
the  product  under  those  figures  of  the  dividend  before  men- 
tioned. 

4.  Subtract  this  product  from  that  part  of  the  dividend 
under  which  it  stands,  and  set  down  the  remainder. 

5.  Bring  down  another  figure  of  the  dividend,  and  place 
it  at  the  right  hand  of  the  remainder,  and  then  divide  as  be- 
fore this  number  so  increased  ;  but  if,  when  one  figure  is 
brought  down,  the  divisor  is  not  contained  in  it,  set  down  a 
cypher  in  the  quotient,  and  bring  down  another  figure  of  the 
dividend  ;  and  so  on,  till  all  the  tigures  of  the  dividend  are 
brought  down  and  divided. 

Eocample,  Divide  561720  by  24. 
Mivis.    Divid.      Quot. 

24)561720(23405         Work.  First,  write  down  the  divi- 

48  dend,   and  at  the  left  hand  of  it,   the 

—  divisor,  separated  from  it  by  a  curve 

81  line,  and  put  another  curve  line  at  the 

72  right,  to  separate  the  quotient. 

—  Next,   consider  how  few  figures  of 

97  the  dividend  will  contain  the  divisor  ; 

96  the  first  one,  5,  will  not,  but  the  two 

— -  first,  56,  will ;  and  24  in  56  is  2  times; 

120  set  down  2  in  the  quotient,  and  multi- 

120  "dIv  the  divisor  bv  it-     2  times  4  ii;  8  ^ 


so  Division. 

set  down  8  under  the  6.  2  times  ^  is  4  ;  set  down  4.  Tlien 
subtract.  8  from  6  I  cannot,  but  8  from  16  leaves  8  ;  set 
down  8.  1  carried  to  4  is  5  ;  5  from  5  leaves  nothing.  The 
remainder  is  8.  To  this  bringdown  the  next  figure  of  the 
dividend,  which  is  I,  and  it  makes  81,  Then  divide  again. 
24  in  8 1  h  3rtimes  ;  set  down  3  in  the  quotient,  and  multiplj. 
S  times  4  is  12 ;  set  down  2  under  the  1,  and  carry  1.  3 
times  4  is  6,  and  1  I  carried  is  7  ;  set  down  7.  Then  subtract. 
2  from  1  1  cannot,  but  2  from  1 1  leaves  9  ;  set  down  9.  1 
carried  to  7  is  8  ;  8  from  8  leaves  0.  The  remainder  is  9, 
to  which  bring  down  7,  and  it  makes  97.  Then  divide  again. 
24  in  97  is  4  times  ;  set  down  4  in  the  quotient,  and  multi- 
ply. 4  times  4  is  16  ;  set  down  6  under  the  7,  and  carry  1. 
4  times  2  is  «,  and  1  I  carried  is  9  ;  set  down  9.  Then  sub- 
tract. 6  from  7  leaves  1  ;  set  down  1.  9  from  9  leaves  0. 
The  remainder  is  I,  to  which  I  bring  down  2,  and  it  makes 
12.  Then  divide  again.  24  in  12  is  0  timeB ;  set  down  0  in 
the  quotient,  and  bring  down  the  next  figure  of  the  dividend, 
which  is  0,  and  annex  it  to  the  12,  and  it  makes  120.  Then 
divide  again.  24  in  ISO  is  5  times  ;  set  down  5  in  the  quo- 
tient, and  multiply.  5  times  4  is  20  ;  set  down  0  under  the 
0,  and  carry  2.  5  times  2  is  10,  and  2  1  carried  is  12  ;  set 
down  12.  This  product  being  equal  to  the  number  from 
which  it  is  to  be  subtracted,  there  is  no  remainder  ;  and  there 
being  no  more  figures  of  the  dividend  to  bring  down,  the 
work  is  finished,  and  the  quotient  is  23405. 

Proof. 
Multiply  the  quotient  by  the  divisor,  and  to  the  product 
add  the  remainder,  if  any  ;  and  the  amount  will  be  equal  to 
the  dividend,  if  the  work  is  right. 

Contractions  in  Division.  \ 

1.  Division  by  a  number  not  exceeding  12,  may  be  expe- 
ditionsly  performed,  by  multiplying  and  subtracting  in  the 
mind,  omitting  to  set  dow  i  the  work,^  excepting  only  the 
quotient,  which  may  be  set  down  immediately  below  the  di- 
vidend. 

2.  When  the  right  hand  figure  or  figures  of  the  divisor  are 
cyphers,  cut  them  off,  and  also  cut  off  as  many  figures  from 
the  light  hand  of  the  dividend;  then  divide  the  remaining 
figures  of  the  dividend  by  the  remaining  Hgures  of  the  divisor. 
If  any  thing  remains  after  this  division,  annex  to  the  right  of 
it  the  figures  eut  off  from  the  dividend,  and  the  whole  will  be 


JDivmon. 


51 


i 


the  true  remainder  ;  if  nothing:  rc^rnains  from  the  division,  the 
iigiuef^  cut  oft'  will  be  the  remainder. 

J\'ote  1.  This  method  is  us(m1  fluly  u)  aYoiU  the  needless  repetition  of  cy- 
phers   which  \v<  aid  haj)pt:n  in  tin-  conmion  'way. 

vVr^i'tf  2.  The  ijuojei  way  of  selling  do"n  a  remainder  after  division,  is  t® 
place  it  a:  tht  i  i^lit  hand  of  the  quotien  ,  with  iJie  divisor  under  it,  and  a 
line  between,  m  the  form  of  a  vulgar  fraction. 


Questions. 

1.  If  there  arc  1189  chapters  in  the  bible,  and  a  child 
should  read  2  chapters  a  day,  how  many  days  would  it  take 
him  CO  read  it  ihrough  ?  Jns,  5 94 J. 

2.  If  a  Cliris'ian  school  for  50  heathen  children,    can  be* 
kept  ill  Ceylon  for  2  00  cents  a  yeai%  according  to  the  state- 
ment of  the  missionaries  there,  how  many  cents  is  that  for 
each  child  ?  dns,  48. 

S.  If  the  population  of  the  United  States  is  9650400,  and 
one  minister  of  the  gospel  is  necessary  for  every  800  souls, 
how  many  ministers  are  necessary  in  the  United  States  ? 

Alls.  12038. 

Questions  on  the  foregoing. 


What  is  division  ? 

When  is  it  called  simple  division  f 

What  is  the  dividend  ?  divisor  P  quo- 
tient ?  [down  ? 

How  ar*^  the  numbers  to  1)6  written 

W  iicav  do    ou  begin  to  divide  ? 

How  many  figures  of  the  dividend 
do  you  take  P 

V/here  do  }Gu  set  your  quotient 
figure  ? 

Wliat  do  }ou  do  next  ? 

From  whnt  do  you  subtract  ? 

What  do  you  do  after  subtracting  ? 

How   many   figures    do   you    bring 


down  at  a  time  i* 
What  do  you  do,  when  the  divisor  is 

not  contained  in  the  remainder  so 

increased  ^ 
How  do  you  prove  division  ? 
What,  is  the  first  method  of  contract- 
ing division  p  the  secoiid  ^ 
In  the  second  method,   what  is  the 

true  remainder  P 
How  should  the  remainder   be  set 

down  P 
What  is  tlie  result  of  an  operation  in 

addition  called  ?    in   subtraction  ? 

in  multiplication  ?  in  division  ? 


Tell  how  many  times 

Ajis, 


5  in 

2^, 

4  in 

20, 

6  in 

3  6, 

Tin 

42, 

8  in 

56, 

8  in 

24, 

5  m 

S7, 

Exercise  12 

4  in    32, 

dns. 

8    9  in 

108, 

3  in    33, 

1  i    6  in 

30, 

5  in  100, 

20    5  in 

45, 

9  in    54, 

6    9  in 

72, 

7  in    m, 

9112  in 

96, 

2  in  120, 

10    9  in 

63, 

5  in    20, 

4  il  in 

99, 

Ms,  12 
5 
9 
8 


3J| 


Exercius* 


Tell  how  many  times 
2  in    4x  4,  Ms.  8 


S  in     :^ 

4  in    2 

5  in  10 
4  in  8 
4  in    6 

6  in    3 


6, 
10, 
4, 
2, 
2, 
8, 


Exi  RCISE   13» 

2  in  5x  6,  Jlns.  V 


3  in  9 
;  in  8 
o  in  4 
12  in  6 
3  in  2 
5  in  3 


6, 

4, 

6, 

15, 

10, 


6 
1 
2 
3 

10 
6 


3  in  6x  4, 

4  in  5  8, 
8  in  4  12, 
6  in  b  4, 
8  in  4  6, 
3  in  2  9, 
i;  m  3  12, 


6 
6 

3 
6 
4 


Tell  how  many  times 

7  in  3x21,     Jins, 

5  in  2    15, 

8  in  3     16, 

6  in  4     12, 
8  in  6     12, 

5  in  4    15, 

6  in  9    10, 


Exercise  14. 


!.  9 

7  in  3x14, 

Ayis,  e 

6  in  2x18, 

Atis,  6 

( 

9  in  4     18, 

8 

4  in  2    14, 

7 

6  3  in  4     12, 

16 

8  in  2    20, 

5 

8[9  in  2     2r, 

6 

7  in  3     2  i , 

9 

9'5  in  3     ^0, 

12 

8  in  4     16, 

8 

12  9  in  3     15, 

5  9  in  2     18, 

4 

15 

8  in  2     16, 

4 

6  in  5     12, 

10 

Tell  how  many  times 

Jtiis, 

6in3x2xS,  3 

8  in  2    2    4,  S 

10  in  2     2     5,  2 

8  in  2    3    4,  8 

6  in  ^^    3    5,  5 


Exercise  l5. 

6  in  3x2x4, 
4  in  3  2  6, 
4  in  2  2  5, 
12  in  6  4  3, 
8  in  5     2    4, 


Ans. 
4 
9 
5 
6 
5 


4  in  5x2x2 
6  in  5  7  2, 
8  in  5  4  4, 
6  in  4  5  3, 
6  in  4    3    3, 


An9. 

5 

7 

10 

10 

6 


12  in  3    4    5,         5|]0in5    5    4,       10112  in  8    2    6,       8 
9  in  3    2    6,         4  12  in  3    2    4,         2110  in  5    6    3, 


Exercise  16, 


Tell  the  sum  of 


7+2  +  4+5. 
9  12  3, 
8 


1, 
2, 
1, 

4, 

6, 


18 
15 

17 
16 
15 
20 
17 


9+2+1+4, 
7     8     3     1, 


2     8 
2     7 


3 
5 
9 
6 


4, 
2, 
3, 
4, 


Am. 
16 
19 
17 
16 
21 
19l4 


5+4+9+2, 
8     3, 

3     2, 


S     5     3,        1419     6 


1, 
2, 
3, 
3, 


Ans, 
20 
19 
16 
20 
If 
18 
25 


Ejcereises, 

3$ 

Exercise  IT. 

Tell  the  sum  of 

Am,                          Jim* 

Am* 

Ans. 

24+33, 

5T 

35  +  51,          96 

66+24, 

901 

44+52, 

96 

75     26, 

101 

45     24,          69 

51      32, 

83 

33     53, 

86 

vj     18, 

45 

34     41,          75 

36     48, 

84 

47     $6, 

73 

>2      13, 

105 

27     49,          76 

24     47,' 

71 

22     34, 

56 

^'9      S4, 

103 

39     28,         67 

55     35, 

90 

54     21, 

75 

\: 

Exercise  18. 

Tell  the 

sum  of 

Ans.                          Alls. 

Ans, 

Am. 

S7+28, 

115 

65+41,        l(/6 

54+55, 

19 

72+4,5, 

iir 

65     4S, 

108 

96     70,       166 

71      36, 

107 

84  ^  47, 

131 

91     42, 

133 

67     36,        103 

83     42, 

125 

69     56, 

125 

82     27, 

109 

78     87,       105 

64      59, 

123 

8.5     b4. 

149 

73     %7, 

no 

62     76»       138 

78     58, 

136 

74     66, 

140 

Exercise  19. 

Tell  how  many  times 

fin  13+ 

15,  Ans.  4 

8  in  19+29,  Ms.  6 

9  in  57+35,  Jm.  8 

6  iu  21 

15,           6 

9  in  i5     23,            4 

7  in  1 1      \7, 

4 

5  in  11 

16,           9 

8  in  45     27,           9 

6  in  33     15, 

8 

4  in  13 

19,           8 

7  in  55     29,          12 

5  in  41      19, 

12 

8  in  27 

£9,           7 

6  in  26     40,          1 1 

7  in  21     f8. 

r 

5  in  21 

14,            7  5  in  13     32,            9 

8  in  17     23, 

5 

7  in  25 

31,            8  8  in  27     13,           5 
Exercise  20. 

4  in  20     32, 

IS 

From    take,    aid  tell  the          Ans.    From    take 

,     and  tell  the 

An9» 

27,        3, 

half,             12 

66,        3 

ninth, 

f 

26,       5, 

third,             7 

73,       7, 

eleventh. 

6 

21,       6, 

fifth,               3 

5^9,       5, 

sixth. 

4 

21,       5, 

fourth,           4 

35.       7, 

fourth, 

7 

36,       6, 

fifth,               6 

43,        8, 

seventh. 

5 

-1,       5, 

third,           12 

52,        4. 

srah, 

3 

51,       9, 

sixth,             7 

57,      15 

six^h. 

7 

62,      14, 

eighth,           6 

63,      \5 

eighth. 

6 

79,       7, 

sixth,           12 

54,       9 

third. 

15 

i^4,       7, 

sey€ 

nth,      1 1 

64,       8 

,        seventh, 

8 

34  ReduGiion. 

REDUCTION, 

Is  the  changing  of  numbers  from  one  name  or  denomina- 
tion to  another,  without  altering  their  value. 

Before  proceeding  to  operations  in  reduction,  the  following 
tables  should  be  committed  to  memory,  as  far  as  Table  15. 

REDUCTION  TABLES. 

1.  Federal  Money. 
10  mills,  (m,)  make         1  cent,  ct. 
10  cents,  1  dime,  d, 

10  dimes,  or  100  cents,    1  dollar,  S,  or  D. 
10  dollars,  1  Eagle,  E. 

JStote,  In  stating  any  sum  in  Federal  Money,  eagles  and  dimes  are  usually 
aeglected,  and  dollars  and  cents  only  arc  mentioned,  100  cents  making  a 
dollar.  I'he  dollar  is  considered  the  money  unit,  and  the  lower  denomina- 
^ons  as  decimal  parts  of  a  dollar » 

2.  Sterling  Money,  and  old  Currencies  of  the  several 
States, 
4  farthings,  (q.)  make       1  penny,  d, 
12  pence,      ^  1  shilling,  s. 

20  shillings,  1  pound,  £,  or  L 

JS^ote.  Farthings  are  usually  written  as  fractional  parts  of  a  penny,  «•  Xd* 
iS  1  farthing ;   jcf.  i»  a  halfpenny,  or  2  farthings ;  %d.  is  f  farthings. 

3.  Troy  Weight. 
24  grains,  (gv,)  make     1  pennyweight,  dwt, 
20  pennyweights,  1  ounce,  ox, 

12  ounces,  1  pound,  Ih, 

J^ote,  A  grain  is  equal  to  *7  ten  thousandths  of  a  solid  inch  of  pure  water. 

This  weight  is  used  for  gold,  silver.  jeAvels,  electuaries,  and  liquors.  The 
Ineness  of  gold  is  tried'hj  fire,  and  is  reckoned  in  carats.  Jf  it  loses  nothing 
in  the  trial,  it  is  said  to  be  Zi  carats  fine.  If  it  loses  2  twenty -fourths,  it  is 
said  to  be  22  carats  fine*  which  is  the  stfmdard  for  gold.  Silver,  which  lose* 
nothing  in  the  fir«,  is  said  to  be  12  ounces  fine.  The  standard  for  silver  coin, 
88  II 02.  tidivt,  of  pure  silver,  and  ISd-wt.  of  copper,  melted  together. 

4.  Avoirdupois  Weight. 
16  drams,  (dr.)  make       1   ounce,  o^, 
16  ounces,  1   pound,  lb. 

28  pounds,  1  quarter,  qr. 

4  quarters,  1  hundred,  Cwt,  or  C, 

20  hundreds,  1  Ton,  T. 

1  quintal  of  fish,  is  equal  to  1  Cwt 


Reduction. 


Jsiote.  This  weight  is  used  for  provisions,  groceries,  hay,  iron,  lead,  and, 
in  general,  for  alT  coarse  and  bulky  articles.  192  ounces  Avoii'dupois  are 
equal  to  175  ounces  Troy  ;  and  l44/ds.  Avoirdupois,  to  175lbs.  Troy;  and 
lib.  Aveirdupois,  to  7000  grains  Troy,  or  to  27-664  solid  inches  of  pure  water, 

5.  Apothecaries'  Weight. 

20  grains,  (gr,)  make         1  scruple,  sc,  or  ^ 

3  scruples,                         1  dram,  dr,  or  31 
8  drams,                             1  ounce,  oz.  or  3^ 

12  ounces,  1  pound,  lb. 

JVote.  This  weight  is  used  for  compounding  medicines.  A  pound  of  this 
weight  is  the  same  as  a  pound  Troy. 

6.  Cloth  Measure. 
2:J  inches,  (in»)  make     1  nail,  na, 

4  nails,  1  quarter,  qr. 

4  quarters,  1  yard,  yd. 

5  quarters,  1  ell  Flemish,  E.  FL 

5  quarters,  1   ell  English,  E.  E. 

6  quarters,  1  ell  Frencb,  E,  Fr, 

7.  Long  Measure. 
6  points,  (pf.j  make      .  1  line,  /. 

4  lines, 

5  barley  corns, 
12  inches, 

3  feet, 
16§  feet,  or  5  j  yds. 

4  rods,  or  66  feet, 
10  chains,  or  40  rods, 

8  furlongs,  or  52  80  ft. 

5  miles, 
69§  statute  miles,(nearly)  1   degree,  deg, 

060  degrees,  I  great  circle  of  the  earth,  cir. 

J^fote.  The  chain  is  divided  into  100  links.  In  measuring  the  height  of 
liorses,  4  inches  make  I  hand  In  measuririj^  depths,  6  feet  make  1  failion. 
Long  measure  is  used  to  measure  distances,  or  any  other  thing  in  which 
length  is  considered  without  regard  to  hreadth. 

8.  Square  Measure. 
144  square  inches,  make     1  square  foot, /f. 
9  feet,  1  yard,   yd, 

30^  yards,  or  %7^\  feet,  '      '" 

40  rods. 


barley  corn,  &.  c. 
1  inch,  in. 
1  foot,/^ 
1  yard,  yd. 

\  rod,pole,orperch,  r.  or|?. 
1  chain,  c/i. 
1  furlong, /wr. 
I  mile,'m. 
1  league,  L. 


4  roods, 

16  rods,  or  iOCOO  links, 
to  chains, 
640  acres, 


I   rod,  pole,  or  perch,  n  or  p. 
1  rood,  R. 
1  acre,  J., 
1  chain,  eh. 
1  acre.  A- 
1  mile,  m 


36  Reduction, 

JVo^e.  This  measure  is  used  to  ascertain  the  quantity  of  aay  thing  whick 
has  length  and  breadth,  without  rega.id  to  tliickness,  as  the  floor  of  a  room, 
the  content  of  a  piece  of  land,  &c.  'I'he  length  and  breadth  being  multiplied 
together,  to  make  the  area,  oi  superficial  content.  Hence,  144  inches  make 
1  foot;  because  \2  inches  in  length,  and  12  inches  in  breadUi,  being  multi- 
plied together,  make  l44  square  inches  in  one  square  foot. 

9.  Solid  or  Cubic  Measure. 


728  solid  inches,  make 

1  solid  foot,  ft 

27  feet. 

1  yard,  yd. 

40  feet  of  round  timber,  or 

50  feet  of  hewn  timber. 

\  ton,  or  load,  T. 

128  feet  of  wood, 

1   cord,  c. 

J^ote.  1  his  measure  is  used  to  ascertain  th^  quantity  of  any  thing  whiek 
has  length,  breadth  and  thickness,  and  to  regulate  all  measures  of  capacity,  (^ 
Tiv^hatever  forni.  A  solid  foot  contains  1728  solid  inches  ;  because  it  is  12 
inches  loi<g.  12  inches  broad,  and  12  mches  thick,  and  12x12x12,  or  tlie  cube 
©f  12,  ia  1728. 

10.  Dry  Measure. 

S3  solid  inches  and  3  fifths,  make  1  pint,  pt, 

2  pints,  ^   quart,  qt, 

4  quarts,  or  268  in.  &  4  fifths,  1  gallon,  ^al. 


8" 


1   peck,  ^k. 


4  pecks,  or  2150  in.  &  2  fifths,  1  bushel,  b. 

8  bus! 5 els,  '  I    quarter,  qr. 

36  bushels,  1   chaldron,  chaU 

Js'ote.  I'liis measure  is  used  for  grain   fruit,  sefeds,  roots,  salt,  ceal,  &c, 

11.   Wine  Measure. 
4  gills,  or  '8  solid  indies,  and 

875  thousandths,  make  1  pint,  ft. 

2  pints,  1    quart,  qt, 

4  quarts,  or  231  inches,  1  gallon,  gal. 

63  gallons,  1  hogshead,  }ihd, 

2  hogsheads,  1  pipe,  p. 

2  pipes,  I  tun,   T. 

JVote.  This  measure  is  used  for  wine,  spirits,  vinegar,  oih  &c. 

12.  Ale,  or  Beer  Measure. 

35^  inches,  make  1  pint,  jpt. 

2  pints,  1   quart,  qt. 

4  quarts,  or  282  inches,    1   gallon,  gal. 

8  gallons,  1   tlrkin  of  ale,  ^. /ir. 

9  gallons,  1  firkin  of  beer,  B.fir. 

2  firkins,  1  kilderkin,  kiL 

5  kildeikins,  1  barrel, 

3  barrels,  1  butt,  Bt. 


Reduction.  3f 

13.  Time. 

60  seconds,  (see,)  make  1  minute,  min. 

60  minutes,  1  hour,  hr. 

24  hours,  1  daj,  d. 

7  days,  1  week,  w. 

4  weeks,  1  lunar  month,  l.  m. 

3654  days,  (nearly,)   or  13  lunar  months,  and  IJ 

days  ;  or  12  solar  months,  1  year,  yr. 

100  years,  1  century,  cenf. 

JSilote.  In  reckoning  timci  365  days  are  usually  considered  a  year*  which  is 
divided  into  12  calendar  months,  as  follows  :  January,  31  days  ;  February, 
28  }  March,  31  ;  April,  30  j  May,  31';  June,  30  ;  July,  31  ;  August,  31  ; 
September,  SO  ;  October,  31 ;  November,  30  ;  Decemberj  31.  The  quarter 
of  a -day  is  reserved  till  it  becomes  a  -whole  day,  which  is  every  fourth  year, 
and  then  it  is  added  to  the  month  of  February,  which  then  has  29  days;  and 
that  year  is  called  leap  y  ear.  To  know  whether  any  year  is  leap  year,  divide 
k  by  4  and  if  there  is  no  remainder,  it  is  leap  year.  But  as  the  true  year  is 
11  minutes  ai»d  12  seconds  less  than  365  days  and  a  quarter,  as  often  as  this 
deficiency  makes  up  a  day^  the  additional  day  for  leap  year  is  omitted  :  which 
is  once  in  about  130  years. 

14.  Circular  Motion. 
60  thirds,  ('")  make         1  second,  '' 
60  seconds,  1  minute,  ' 

60  minutes,  1  degree,   ° 

SO  degrees,  1  sign,  S. 

12  signs,  or  360  deg.        1  circle,  or  complete  re- 
volution of  the  Zodiac. 

115.  Particulars. 
1 2  particulars,  make     1  dozen,  doz. 
20  particulars,  1  score,  sc. 

12  dozen,  1  gross,  gr, 

12  gross,  1  great  g'rosS,  g,  gr. 

16.  Coins  and  Currencies. 
TTie  Spanish  or  Federal  dollar  is  equal  to 
I      4s.     6d,  Sterling  money  of  Great  Britain  ; 
4..  10 J   Irish; 

5  ..    0  Canada  and  Nova-Scotia  ; 

6  ..   0  New-England,  Virginia,  Kentucky,  and  Tennessee  ; 
8 ..    0  New-York  and  North- Carolina ; 
7 ..    6  Pennsylvania,  N.  Jersey,  Delaware,  and  Maryland  ; 
4..    8  South-Carolina  and  Georgia; 

5  livres  and  2  fifths,  of  France  ; 

2  guilders,  or  florins,  and  4  sevenths,  of  the  Netherlands  ; 

3  marcs  banco  of  Hamburgh  ; 
1  rix  dollar  of  Denmark,  Sweden  and  Hamburgh ; 

10  rials  of  plate,  or  ^0  rials  of  vellon,  of  Spain. 


3^.  Reductiofu 

Gold  Coins, 
Johannes,  equal  to      8 1 6-00,    weighs  ISdwt  Ogr. 
Doubloon,  14'933, 

Moidore,  6-00, 

English  guinea^  4'66r, 

French  guinea,  4'6o, 

Spanish  pistole,  3*773, 

French  pistole,  3' 667, 

Louis  d'or,  4'444, 

Silver  Coins, 
English  or  French  crown,   gMo,  weighs  19dwt.  Ogr, 
Spanish  dollar,  i'OO,  17..     6 

English  shilling,  -222,  3  ..   18 

Pistareen,  •20,  3  ..    11 

All  other  gold  coins  of  equal  fineness,  at  89  cents  per 
dwt,  and  silver  at  U 1  cents  per  oz.    An  English  guinea  is 
lisuallj  reckoned  21s.  sterling,  and  a  crown  5s, 
Other  Currencies, 


16  .. 

21 

6  .. 

18 

5  .. 

6 

5   .. 

5 

4  .. 

6 

4  .. 

4 

Ruble  of  Russia, 

m55 

Franc  of  France, 

•1813 

Mill  ree  of  Portugal, 

1-24 

Tale  of  China, 

1-48 

Sequin  of  Arabia, 

1-64 

Pagoda  of  India, 
Rupee  of  Beqgal, 
Piastre  of  Turkey, 
Ducal  banco  of  Venice, 

^1-84 
•50 
•888  + 
•926 

Ducat  of  Vienna, 

2-055  + 

But  these  often  vary  according  to  the  rate  of  exchange. 

17.  Ancient  Weights,  Coins,  and  Measures. 

Hebrew,  Greek,  and  Roman  drachma  or  dram,  equals  Qdivt  &^rs.  and  S 
fourtlis,  Troy  weight.  Dram  of  silver,  6  pence  and  i7  thirty-seconds  ster- 
ling, or  12  cents  and  97  one  hundred  and  forty -fourths.  Dram  of  gold,  95. 
Id,  2q.  sterling,  or  2  doUs.  2  cts.  and  7  ninths.  Shekel  of  silver,  brass,  iron, 
&c.  4  drachma.  Shekel  of  gold,  2  drachmae.  5  gerahs  make  I  dracfima. 
Mina  or  pound,  60  shekels.  Hebrew  talent,  12000  drachmse  Attic  mina, 
100  drachmae.  Smaller  Attic  talent,  6000  drachmsc.  Greater  Attic  talent, 
8000  drachmse.  Stater  of  silver,  4  drachm ge.  Stater  of  gold,  2  ^rachmse. 
7  lepta,  or  mites,  1  chalcus ;  8  chalci,  1  obelus  5  6  oboli,  1  drachma.  Roman 
ounce,  8  drachmae  5  pound,  96  drachmse.  4  terentii,  or  quadrantes,  1  as  ;  2 
and  a  half  asses,  1  sestertius,  ornummus;  4  sestertii,  1  denarius,  penny,  or 
drachma.  Sestertium,  1000  sestertii.  Roman  talent,  24  sestertia,  or  600© 
drachmse. 

JVote,  The  as  is  often  expressed,  in  the  Latin  classics,  by  L.  for  libra,  a 
pound,  because  it  was  originally  a  pomnd  of  brass.  The  sestertius,  by  LLS, 
(corrupted  into  HS,)  two  pounds  and  a  half.  When  a  genitive  plural  is  used 
after  a  numeral  adjective,  it  denotes  so  many  thousands ;  and  when  after  9t 
numeral  adverb,  so  many  hundred  thousands. 

Long  Measure,  Hebrexv.  4  fingers'  breadth,  1  band  bj^eadtb;  2  hand 
breadths,  I  shorter  spjm  ;  S  hand  breadths,  1  longer  span  ;  2  longer  spans,  1 
cubit,  equal  to  1-824  English  feet;  4  cubits,  1  fathom;  6  cubits,  1  Ezekiel'a 
reed  ;  80  cubits,  1  stihoenus,  or  measuring  line ;  667*5  English  feet,  1  sta- 
dium, or  furlong ;  8  stadia,  1  mile;  30  stadia^  1  pairaMng;  240  stadia,  I 
iay's  journey  5  5  sta,dia>  1  SlabbatK day's  journey, 


Reduction. 


^reek,  4  fingers,  1  doron ;  4  dora,  1  foot,  equal  to  120875  English  inches ; 
18  fingers,  1  pugme.  or  smaller  cubit ;  24  fingers,  1  pechus,  or  larger  cubit  ; 
4  larger  cubits,  1  orguia,  or  pace  ;  100  paces,  1  stadium ;  8  stadia,  1  mile. 

Roman,  4  fingers,  I  palmus  minor;  4  pa'mi,  1  foot,  equal  to  11-604  En- 
glish inches ;  24  fingers,  1  cubit ;  40  fingers,  1  gradus ;  5  feet,  1  pass\is  j 
625  feet,  1  stadium;  lOOO  passus,  1  mile. 

Square  Measure.  Greek  or  Egyptian  aroura*  10000  square  cubits  ;  Greek 
plethron,  2  arour«  ;  Roman  juger'am,  2  English  roods,  10  poles,  and  25005 
ieet. 

Cubic  Measure.  Hebrew,  10  avoirdupois  ounces  of  rain  water,  1  cotjift; 
10  cotylse.  1  gomer,  or  omer  ;  10  omers,  1  bath,  epbst,  or  metretes,  equal  to 
6®  English  ^vine  pints,  and  15  solid  inches  ;  3  baths,  1  rebel ;  5  baths*  1  le- 
theck;  10  baths,  1  cor,  or  homer.  6  eggs^  or  betzahs,  1  log,  or  rebah  ;  4 
logs.  1  cab  ;  3  cabs.  1  hin  ;  2  bins,  I  seah ;  3  seahs,  1  bath. 

Greeks  for  liquids.  5  cochlearia,  1  concha;  2  conchse,  1  cyathns;  6  cyathi. 
1  cotyle  ;  2  cotylse,  1  xestes ;  6  xest^,  1  chous  ;  12  choi,  1  metretes,  equal  to 
82  English  wine  pints,  and  19-626  Solid  inches.  For  things  dry»  3  cotylae,  1 
choenix  ;  48  chcenices,  1  medimnus,  equal  to  4  English  pecks,  S[nd  205-101 
solid  inches. 

Romany  for  liquids.  41igulse.  1  cyathus;  12  cyathi,  1  sextarius;  i  sex- 
tarii,  1  congius  ;  4  congii,  1  urna ;  2  urnce,  1  amphora,  equal  to  57  English 
wine  pints,  and  10* 66  solid  mcheis;  20  amphorse,  1  culeus.  For  things  dry, 
16  sextarii,  1  modius,  equal  to  1  English  peck,  and  7-68  Solid  inches. 

18.  Specific  Gratities, 

The  following  table,  (taken  chiefly  from  Enfield's  Philosophy,)  shows  the 
weight  in  avoirdupois  ounces  of  a  solid  Foot  of  each  substance. 


Platina,  (pure,) 
Fine  s:old. 
Standard  gold, 
Mercur}^, 
Lead, 
Fine  silver. 
Standard  silver. 
Copper, 
Gun  metal? 
!Fine  brass. 
Steel, 
Ii'on, 
Pewter, 
Cast  iron, 
Load  stot^e. 
Diamond, 


23000 

19640 

18888 

14019 

11325 

11091 

10535 

9000 

8784 

8350 

7850 

7645 

7471 

7425 

4930 

3517 


White  lead, 

Marble, 

Rock  chrystal. 

Greet!  glass, 

Flint  stone. 

Brick, 

Ivory, 

Sulphur, 

Chalk, 

Alum, 

Lignum  vitfe, 

Coal, 

Ebony, 

Mahogany, 

Cows'  milk. 

Boxwood, 


■oz.. 

3160 

2705 

2658 

2600 

2570 

2000 

1825 

1810 

1793 

1714 

1327 

1250 

1117 

1063 

1034 

1030 


GZ, 

Sea  water, 

103d 

Rain  water, 

1000 

Red  wine, 

993 

Proof  spirits, 

925 

Dry  oak. 

925 

Olive  oil. 

913 

Ice, 

908 

Living  men, 

891 

Spts.  ofturpentine,864 

Alcohol, 

850 

Elm  and  ash, 

800 

Ether, 

732 

White  pine. 

569 

Cork, 

240 

Common  air. 

114 

Inflammable  air, 

•12 

19.  Miles  of  different  Countries* 

11  Irish  equal  14  English ;  I  Scotch  equals  I  and  a  half  English  •  I  Indian 
equals  3  Fit^glish  ;  I  Dutch,  Spanish  and  Polish,  equals  3  and  a  half  English  ; 
1  German  equals  4  English ;  I  Swedish,  Danish,  and  Hungarian,  equals  5 
and  a  lialfEnglish;  I  Russian  verst  equals  3  quarters  of  a  mile  English, 

(j^  Sound  moves  1142  feet  in  one  second  of  time. 

Light  flies  from  the  sun  to  the  earth,  which  is  nearly  94  millions  of  miles, 
in  about  8  minutes  of  time,  which  is  nearly  195834  miles  per  second  ;  so 
that  for  any  short  distance  it  may  be  considered  instantaneous. 


40  Reduction, 

REDUCTION, 
Is  performed  by  multiplication  and  division. 

Rule. 

1.  When  the  given  number  is  to  be  reduced  from  a  higher 
denomination  to  a  lower,  multiply  the  given  number  of  the 
highest  denomination  by  as  many  units  as  it  takes  of  the 
next  lower  to  make  one  of  that  higher  ;  to  this  product  add 
the  number,  if  any,  which  was  in  this  lower  denomination 
before,  and  set  down  the  amount.  And  so  on,  through  all 
the  denominations,  till  you  have  brought  the  number  into  the 
denomination  required. 

2.  When  the  given  number  is  to  be  reduced  from  a  lower 
denomination  to  a  kigher,  divide  the  given  number  by  as 
many  units  as  it  takes  of  that  denomination  to  make  one  of 
the  next  higher,  and  set  down  what  remains,  as  well  as  the 
quotient.  And  so  on,  through  all  the  denominations,  till  you 
have  brought  the  given  number  into  the  denomination  re= 
quired. 

Example  1. 

Reduce  ^1234 ..  15  ..  7,  to  farthings. 

Operation.  Eocplanation.  Here,  because   20  shil- 

L        s,    d.    lings  make  1  pound,  I  multiply  the  1 234 

1234  ..  15  ..  7    pounds  by  20,  to  bring  them  into  shillings, 

20  and  have  S4680  shilling;*,  to  which  1  add 

the  15  shillings  of  the  given  number,  and 

have  24695  shillings.     Then,  because  12 
pence   make    1    shilling,   I  multiply  the 
24695  shillings  by  12,  to  bring  them  into 
24695  s.  pence,  and  have  296540  pence,  to  which 

12  I  add  the  7  pence  of  the  given  number, 

and  have  296347  pence.  Then,  because 
4  farthings  make  1  penny,  1  multiply  the 
296347  pence  by  4,   to  bring  them  into 

farthings,  and  have  1185388  farthings  ; 

296347  d,  and  as  there  were  no  farthings   in  the 

4  given  number,  1  have  nothing  to  add,  and 

the  answer  is  1 185388  farthings. 

1185388  q*  Answer. 


Reduction, 


41 


Example  2. 

Reduce  1185388  farthings  to  pounds. 
Operation.  —     - 

4)1185388  q. 

12)296347  d. 


dei2S4..  15, 
dnswer. 


Explanation.  Here,  because  4  farthings 

make  1  penny,  I  divide  the  given  farthings 

by  4,  to  bring  them  into  pence,  and  have 

296347  pence.    Then,  because  ISpence 

make  1  shilling,  I  divide  the  pence  by  12, 

20)24695  s.  7d.  to  bring  them  into  shillings,  and  have 

24695   shillings,    and  7  pence  remains, 

7,    which  I  set  down.     Then,  because  20 

shillings  make  1  pound,  I  divide  the 

shillings  by  20,   to  bring  them  into 

pounds,  and  have  1234  pounds,  and  15  shillings  remains. 

So  the  answer  is  1234Z.  15s.  7rf. 

Questions. 

1.  In  the  year  1819,  the  receipts  of  the  British  and  Fo- 
reign Bible  Society  were  9S0SSL  6s.  Td.,  how  many  farthings 
was  that  ?  Ms.  893 1 1996. 

2.  The  time  from  the  creation  to  the  flood  was  14  cen- 
turies and  56  years  ;  how  many  calendar  months  was  it  ? 

Ms.  17472. 

S.  The  highest  mountain  in  the  known  world  is  Himalaya, 
in  India,  which  is  estimated  to  be  952632  barley  corns  above 
the  level  of  the  sea ;  liow  many  miles  is  that  ?  Ms.  5m.  62/f. 

4.  The  next  is  Chimborazo,  in  South  America,  which  is 
estimated  to  be  4m.  330/f.  ^  how  many  inches  is  that  ? 

/  ^«s.  257400. 


Exercise  2^1. 

the  pence  in 

7s.     Sd. 

Ans.  87 

Ss. 

Sd. 

Ms.  101 

14..    2, 

170 

9.. 

9, 

117 

15..    6, 

186 

11.. 

11, 

143 

13..    4, 

160 

13.. 

6, 

162 

5..  11, 

71 

10.. 

5, 

125 

10..  10, 

130 

16.. 

0, 

192 

9..    8, 

116 

12.. 

6, 

150 

r..  6, 

90 

21.. 

0, 

252 

^•.  10, 

82 

H.. 

8. 

17^ 

D2 


42 

Eeduction. 

Exercise  23. 

Tell  the  farthingj 

3  in 

2s.     Sd. 

^n5.  176 

SS. 

6cZ. 

^ns.  168 

2..    5, 

116 

2.. 

3, 

108 

4..    4, 

208 

4.. 

3, 

204 

r..  6, 

360 

2.. 

1, 

100 

6..    2, 

S96 

3.. 

4, 

160 

5..    6. 

264 

4.. 

li. 

198 

3..    6, 

164 

1.. 

u. 

53 

2..    6, 

120 

3.. 

3^, 

158 

4..    5, 

212 

2.. 

4i. 

114 

COMPOUND  ADDITION, 

Is  the  addition  of  several  numbers  of  difterent  denomina- 
tions, but  of  the  same  general  nature. 

Rule. 

1.  Write  down  the  numbers  in  such  a  manner  that  those 
of  the  same  denomination  may  stand  directly  under  each 
other,  and  the  lowest  denomination  at  the  right  hand,  the 
next  lowest  next,  and  so  on  ;  and  draw  a  line  underneath. 

2.  Add  up  the  numbers  of  the  right  hand  column,  as  in 
simple  addition  ;  and  find,  by  reduction,  how  many  units  of 
the  next  higher  denomination  are  contained  in  their  sum. 
Set  down  the  remainder  under  that  column,  andxarry  those 
units  to  the  next  column. 

3.  Proceed  in  the  same  manner  through  the  several  deno- 
mination-, to  the  highest,  the  sum  of  which,  together  with  the 
several  remainders,  at  the  foot  of  the  other  columns,  will  be 
the  answer  sought. 

"  Work,  Begin  with  the  right  hand  co- 

lumn, at  the  bottom,  and  gay,  15  and  11 
is  26,  and  10  is  36,  and  9  is  45,  and  8  is 
53.  This  is  53  ounces;  and  as  16 ounces 
make  1  pound,  divide  53  by  16.  1 6  in 
53  is  3  times,  and  6  remains.  Set  down 
5  ounces,  and  carry  3  to  the  columrs  of 
pounds.  Second  column.  3  carried  to  1 3 
IS  1 6,  and  If  is  33,  and  27  is  60,  and  21 
is  81,  and  15  is  96.  This  is  96  pounds  ; 
and  as  2Slb.  make  1  quarter,   diyide  96 

by  28.    28  in  96  is  3  times,  and  ;  2  remains.  Set  down  \2lb. 

and  carry  3  to  the  next.     Third  column,  3  carried  to  1  is  4, 


Example. 

Cwt  qr.  lb. 

oz. 

S..S..15. 

.    8 

4..2..21. 

.    9 

5..  1..27. 

.10 

4..3..17. 

.  11 

5..  1..  13. 

.  15 

21..  1..  12. 

.    5 

Answer. 

Compound  Addition*  43 

and  3  is  7,  and  l  is  8,  and  2  is  10,  and  3  is  15.  This  is  13 
quarters;  and  as  4  quarters  make  i  Cwt.  divide  13  by  4. 
4  in  13  is  3  times,  and  1  remains.  Set  down  \qr.  and  carry 
3  to  the  next.  Fourth  column,  3  carried  to  5  is  8,  and  4  is 
12,  and  3  is  15,  and  4  is  i9,  and  3  is  2^Z.  This  is  22  Cwt,  ; 
and  as  this  is  the  last  column,  set  down  22  Cwt.  And  the 
answer  is  22  Cwt.  \qr.  I'zlh,  Soz, 

Questions. 

1.  The  government  expenses  of  the  United  States  for  the 
year  1819,  were  estimated  ab  follows:  Civil,  diplomatic,  and 
miscellaneous,  D.l6l9836vn  ;  military  and  Indian,  D. 8666- 
252'8.5 ;  naval,  D.3802485*60  ;  public  buildings,  and  roads, 

.  D.326644  ;  public  debt,  D.lOOOOO^iO ;  erecting  custom  hous- 
es, &c.  D.  100000  :  what  is  the  amount  ? 

^ws.  24515219-76. 

2.  The  receipts  of  the  B.  and  F.  Bible  Society  for  the  first 
eleven  years,  were  as  follows  :  First,  5592L  J  Os.  5d, ;  se- 
cond, 8827L  iOs.  Sid, ;  third,  6998L  19s.  Td.;  fourth,  10039/. 
12s.  O^d.;  fifth,  11289Z.  I5s.  3i.;  sixth,  23337^  Os.  ^id  ; 
seventh,  25998Z.  35.  Id.;  eighth.  4353^Z.  12s.  5^d^, ;  ninth, 
76455/.  Is.;  tenths  87216/.  6s.  9d.;  eleventh,  99894/.  15s. 
6d.  :  how  much  in  all  ?  Am.  399182/.  6s.  7d. 

3.  Tell  the  whole  weight  of  the  following  parcels  of  me- 
dicine, to  wit :  First,  3/6.  5o%.  7dr.  2sc. ;  second,  6oz.  5dr. 
isc.  l6gr. ;  third,  5/6.  loz.  6dr.  2sc.  lOgr. ;  and  the  fourth, 
9oz.  3dr.  19gr.  Ana.  9/6.  l\oz.  7 dr.  Isc.  5gr. 

4.  Add  the  following  distances,  and  tell  the  amount :  first, 
I27yd.  I  ft.  Sin.  26.  c. ;  second,  \2yd.  \Oin.  16.  c. ;  third,  2ft. 
II in. ;  fourth,  9yd.  Tin.  26.  c. ;  and  fifrh,  I2yd.  9.ft.  16.  c. 

Ans.  ]62yd.  iff.  Win. 

5.  Five  vessels  of  wine  contain  as  follows :  first,  6 1  gal. 
2qt. ;  second,  30  gal.  Iqt.  Ipt.  Sg. ;  third,  48  gal.  Sqt.  Sg.  ; 
fourth,  57  gal.  \pt.;  fifth,  i  hlid,  60  gal.  Ipt.Sg.:  what  is 
the  whole  quantity  ?  Ans.  5  hhd.  6  gal.  Ipt.  Ig. 


COMPOUND  SUBTRACTION, 

Is  the  subtraction  of  one  number  from  another,  when  those 
numbers  are  made  up  of  different  denominations,  but  of  the 
same  general  nature. 

Rule. 

1.  Set  down  the  less  number  under  the  greater,  in  suoh  a 
laam^r  that  those  parts  which  are  of  the  same  denominati©n 


44  Compound  Subtraction. 

may  stand  under  each  other,  and  the  lower  denomination  at 
the  right  hand  of  the  higher. 

2.  Begin  at  the  right  hand,  and  subtract  each  part  in  the 
lower  line  from  that  above,  and  set  down  the  difference. 

3.  But  if  any  part  in  the  lower  line  is  gi-eater  than  that 
above  it,  borrow  as  many  of  that  denomination  as  makes  one 
of  the  next  higher,  and  add  to  the  upper  part,  and  then  sub- 
tract the  lower  part  from  the  upper  one  thus  increased,  and 
set  down  the  remainder. 

4.  Carry  1  to  the  next  higher  denomination  in  the  lower    / 
line,   as  an  equivalent  to  what  you  borrowed  in  the  upper,    I 
and  proceed  as  before  ;  and  so  on,  till  the  whole  is  finished,   f 
Then  the  sereral  remainders,  taken  together,  will  be  the 
whole  difference  sought. 

Example.  Work*  Begin  with  the  farthings  at 

L  s.  d,  the  right  hand,  and  say,  a  halfpenny,  or 
From  345..  19..  6|  2  far  Slings  from  3  tarthings,  leaves 
Take     97..  i0..8j     one  farthing,   or  i  of  a  penny.     Set    n 

down  ^flf.    Pence.  8  from  6  I  cannot,    (j 

£48 ..    8  ..  10 J    As  12  pence  make  1  shilling,  I  bofrow 
•Answer.  12  and  add  to  the  6,  and  it  makes  18; 

8  from  18  leaves  10.  Set  down  10, 
and  carry  1.  Shillings,  Having  added  12  pence,  or  1  shil- 
ling, to  the  upper  line  in  the  column  of  pence,  I  must  now 
carry  or  add  1  shilling  to  the  low^er  line  in  the  columa  of 
shillings,  as  an  equivalent.  1  to  10  is  11  ;  11  from  19 
leaves  8.  Set  down  8.  Pounds.  7  from  5  1  cannot.  This 
being  the  last  and  highest  denomination,  I  cannot  borrow  as 
many  as  makes  one  higlier,  but  must  proceed  as  in  simple 
subtraction.  Borrow  10,  and  add  to  the  5,  and  it  makes  15  ; 
7  from  15  leaves  8.  Set  down  8,  and  carry  1.  1  to  9  i*  10  ; 
10  from  4  I  cannot.  Borrow  10,  and  loTrom  14  leaves  4. 
Set  down  4,  and  carry  1,  1  t6  0  is  1  ;  I  from  3  leaves  2. 
Set  down  2.  And  the  answer  is  248^  8s.  lOid. 
Questions. 

1.  The  total  expenditure  of  the  B.  and  F.  Bible  Society 
for  the  first  seventeen  years,  was  908248L  10s.  6^^.,  of  which 
828687/.  17s.  were  expended  in  the  first  sixteen  years ;  what 
was  the  expenditure  of  the  seventeenth  ? 

dns.  79560Z.  13s.  6d. 

2.  The  Baptist  missionaries  at  §erampore  had  expended 
in  translating  and  printing  the  scriptures,  from  1799  to  1809, 
S36445-7>^,  and  had  received  for  that  purpose,  S39574-17: 
what  was  then  unexpended  ?  dns.  S3128'45o 


Compound  Multiplication.  45 

3.  A  jeweller  bought  7lb.  Soz.  14dwt»  ll^r.  of  silver,  and 
made  up  into  spoons  Sib.  7 ox.  ISdwt.  19gr. ;  how  much  was 
left?  Jtns.  3lb.  Toz.  IBdwt.  Ugr. 

4.  Bought  5  Cwt,  \7lb.  5oz.  Gc^*".  of  sugar,  and  sold  3  Cwt. 
2qr.  2llb.  3oz,  IQdr, ;  how  much  was  left  ? 

Ms.  1  Cwt.  Iqr,  24lb.  loz.  I2dr. 


COMPOUND  MULTIPLICATION, 

Is  when  the  multiplicand  consists  of  different  denomina- 
tions. 

Rule  1. 

1.  Set  the  multiplier  under  the  lowest  denomination  of  the 
multiplicand. 

2.  Multiply  the  lowest  denomination  of  the  multiplicand 
by  the  multiplier ;  find  how  many  units  of  the  next  higher 
denomination  are  contained  in  that  product,  set  down  the 
remainder,  and  carry  those  units  to  the  product  of  the  next 
denomination. 

3.  Multiply  the  next  denomination  of  the  multiplicand  by 
the  multiplier,  add  to  that  product  what  was  carried  from  the 
product  of  the  denomination  below,  find  how  many  units  of 
the  next  higher  denomination  are  contained  in  this  product 
so  increased,  set  down  the  remainder,  and  carry  those  units 
to  the  product  of  the  next  denomination. 

4.  Proceed  in  this  manner  to  the  highest  denomination, 
and  set  down  the  whole  of  that  product,  which,  together  with 
the  several  remainders,  will  be  the  answer. 

Work.  Begin  v/ith  the  farthings,  at 
the  right  hand  column,  and  say,  7 
times  1  farthing  is  7  farfliings,  which 
is  1  penny  and  3  farthings.  Set 
down  3,  and  carry  1  to  the  pence. 
Prod.  250  ..  10 ..  4|  Pence.  7  times  9  is  63y  and  1  I  car- 
ried is  64.  64  pence  is  5  shillings 
and  4  pence.  Set  down  4,  and  cany  5.  Shillings.  7  times 
15  is  105,  and  5  1  carried  is  IK*.  110  shillings  is  5  po*  nds 
and  10  shillings.  Set  down  10,  and  carry  5.  Pounds.  7 
times  5  is  35,  and  'J  I  carried  is  40.  Set  down  0,  and  carry 
4.  7  times  3  is  21,  and  4  I  carried  is  25.  Set  down  23  1 
and  the  answer  is  250^.  los.  4%d. 


Example  1. 

/. 

s. 

d. 

Mul 

tiply 

35.. 

.15. 

..QA 

By 

7 

46 


Compound  Divismi. 


Example  2. 
Ih.  oz.  dwt  gr. 
Mult.  21..  1..  7..  13 
By  4 


Prod.84..  5..  10 ..4 


Example  3. 
M.fiir.    p.  yd.  ft 
24  ..  5  ..  20  ..  4  ..  2 
5 


1£2..  1..24..  1..  1 


Example  4. 
el.    hr.  min,  see. 
.6..  20..  35..  51 
6 


23..  6  ..3..  35, 


Rule  2. 

Reduce  the  multiplicand  to  the  lowest  denomination  of 
wliich  it  consists,  and  then  multiply  as  in  simple  multiplica- 
tion ;  reducing  the  product,  when  found,  to  a  higher  deno- 
mination, if  required. 

Example. 

Multiply  35L  15s.  9id.  by  7. 

Work,  I  first  reduce  the  35^  15s.  9 Id,  to  farthings,  and 
find  it  to  be  34357  farthings.  I  then  multiply  those  34357 
farthings  by  7,  and  the  product  is  940499  farthings.  I  then 
reduce  these  240499  farthings  of  the  product  to  pounds,  and 
have  250Z.  10s.  4Jci.  for  the  answer,  as  in  the  first  example. 


COMPOUND  DIVISION, 

Is  when  the  dividend  consists  of  different  denominations. 
Rule  1. 

1.  Write  down  the  dividend  and  divisor  as  in  simple  di- 
vision. 

2.  Find  how  many  times  the  divisor  is  contained  in  the 
highest  denomination  of  the  dividend,  and  put  that  amount 
in  the  quotient,  as  a  part  of  the  answer  of  the  same  denomi- 
nation. 

3.  If  there  is  any  remainder  after  the  division  of  the  high- 
est denomination,  reduce  that  remainder  to  the  next  lower 
denomination,  and  add  to  it  the  number  (if  any)  which  is  al- 
ready in  that  denomination. 

4.  Divide  again,  as  before,  and  so  on,  till  the  last  denomi- 
nation has  been  divided ;  and  the  several  numbers  of  the 
quotient,  taken  together,  will  be  the  answer. 


Compound  Division. 


47 


Example  1. 
Sd.  by  16. 

Explanation.  I  begin  with  the  pounds, 
and  say,  16  in  SO  is  once.  Set  down  1 
in  the  quotient,  in  the  place  of  pounds. 
After  multiplying  and  subtracting,  as  in. 
simple  division,  1  find  14  remains.  This 
being  14  pounds,  is  to  be  reduced  to 
shillings;  and  there  being 20  shillings  in 
a  pound,  I  multiply  14  by  20,  and  the 
product  i3  280,  which  is  280  shillings.  I 
then  add  the  18  shillings  of  the  dividend, 
and  it  makes  298  shillings.  I  then  di- 
vide again.  l6  in  29  is  once,  and  13  re- 
mains. Set  down  1  in  the  quotient,  ia 
the  place  of  shillings,  and  bring  down 
the  8.  16  in  138  is  8  times,  and  10  re- 
mains. Set  down  8  in  the  quotient,  also 
in  the  place  of  shillings,  which  makes  18 
shillings.  The  remainder,  being  10  shil- 
lings, is  to  be  reduced  to  pence;  and 
there  being  12  pence  in  a  shilling,  I 
multiply  10  by  12,  and  the  product  is 
120  pence.  I  then  add  the  8  pence  of 
the  dividend,  and  it  makes  1 28  pence. 
I  then  divide  again.  16  in  128  is  8  times. 
Set  down  8  in  the  quotient,  in  the  place 
of  pence.  After  multiplying  and  subtracting,  I  find  there  is 
no  remainder,  and  I  have  divided  the  last  denomination.  So 
the  answer  is  l^.  I8s.  Bd. 


Divide  30L  18s. 
Operation. 
16)30..  18..  8(U. 
16 

14 
20 

280 
18 

298(1 8s. 
16 

138 
128 

10 
12 

120 
8 

128(8d. 
128 


Example  2. 

lb.   oz.  dwt.gr. 
J>rJ)Z3"7'.G"12,Bd. 


Q$,      3 ..  4 ..  9 ..  12 


Example  3. 
lb.  ox.  dr.  scgr. 
12)13"  1- 2  "O-O 

1"  1..0-2-10 

Rule  2. 


Example  4. 
yd.    qr.  na^ 
47)571  ..  2  ..  1 

12..  0..2-(- 


Reduce  the  dividend  to  the  lowest  denomination  of  which 
it  consists,  and  then  divide  as  in  simple  division  ;  reducing 
the  answer,  w^hen  found,  to  a  higher  denomination,  if  re^ 
quired. 


48 


Compound  Division, 


Example. 

Divide  30?.  18s.  8(Z.  hy  16. 

Work.  I  first  reduce  the  30^.  1 8s.  Sd,  to  pence,  and  find  it 
to  be  7424  pence.  I  then  divide  7424  by  1 6,  and  find  the 
quotient  to  be  464,  which  is  464  pence.  I  then  reduce  464 
pence  to  pounds,  and  have  1^.  IBs.  Sd.  for  the  answer. 

Questions  on  the  fouegoing. 


"What  is  Reduction  ? 

By  what  other  rules  is  it  performed  ? 

\Vhen  the  reduction  is  from  a  high- 
er denomination  to  a  lower,  how  is 
it  performed  ? 

"When  from  a  lower  to  a  higher,  how 
is  it  pei  foraied  ? 

What  is  Compound  Addition  ? 

How  does  it  differ  from  Simple  Ad- 
dition ? 

How  are  the  numbers  to  be  written 
down  ? 

Which  column  is  to  be  added  first  ? 

What  is  to  be  done  with  its  sum  r 

For  how  miany  of  one  denomination 
must  you  carrj'^  1  to  the  next  high- 
er ? 

What  is  Compound  Subtraction  ? 

How  does  it  differ  from  Simple  Sub- 
traction ? 

How  are  the  numbers  to  be  written 
down  ? 

What  is  to  be  done  when  the  lower 


number  is  greater  than  that  above 
it? 

What  is  Compound  Multiplication  ? 

How  does  it  differ  from  Simple  Mul- 
tiplication ? 

How  must  the  numbers  be  placed  ? 

W^here  do  you  begin  to  multiply  f 

What  must  be  done  with  tliat  pro- 
duct ? 

For  how  many  in  the  product  of  a 
lower  denomination  do  you  carry 
1  to  the  product  of  a  higher  .' 

What  is  the  second  rule  »' 

What  is  Compound  Division  ? 

How  does  it  differ  from  Simple  Di- 
vision ? 

How  are  the  dividend  and  divisor  to 
bt  placed  f 

Where  do  you  be^in  to  divide  ? 

When  you  have  divided  «  higher  de- 
nomination, what  do  you  do  v/ith 
the  remainder  ? 

W^hat  do  you  do  next  ? 

What  is  the  second  rule  ? 


Exercise  23. 

From 

take. 

and  tell  the 

^ns. 

From 

take» 

and  tell  the 

Jlns. 

97, 

25, 

twelfth. 

6 

87, 

15, 

eighth. 

9 

101, 

26, 

fifteenth. 

5 

112, 

4o, 

third, 

22 

89, 

26, 

ninth, 

7 

28, 

7, 

seventh. 

3 

67, 

12, 

fifth. 

11 

89, 

25, 

eighth. 

8 

53, 

13, 

fourth. 

10 

27, 

3, 

eighth. 

3 

65, 

11, 

sixth. 

•    9 

38, 

11, 

third. 

9 

74, 

11, 

ninth. 

7 

44, 

8, 

sixth. 

6 

29, 

2, 

third. 

9 

57, 

12, 

third. 

15 

36, 

8, 

fourth. 

7 

48, 

13, 

fifth. 

7 

103, 

49, 

sixth. 

9 

59, 

11, 

sixth. 

8 

96, 

12, 

twelfth. 

7 

63, 

9, 

ninth. 

6 

' 

Ea^ercises. 

49 

EXERC] 

[SE  24. 

Tell  what  is  the 

Jtns. 

Tell  what  is  the 

Am* 

Sd  of  a  half  of  18, 

3 

3d  of  a  5th  of  45, 

S 

4th  of  a  3d  of  24, 

2 

5th  of  a  half  of  50, 

5 

3d  of  a  half  of  36, 

6 

5th  ofa  half  of  20, 

2 

half  of  a  3d  of  24, 

4 

3dof  a6thof  36, 

2 

3d  of  a  4th  of  48, 

4 

4th  of  a  5th  of  40, 

2 

4th  of  a  half  of  40, 

5 

4th  ofa  3d  of  48, 

4 

Sd  of  a  3d  of  27, 

S 

Sd  ofa  1 2th  of 72, 

2 

4th  of  a  3d  of  36, 

3 

half  of  a  12th  of  96, 

4 

3d  ofa  4th  of  60, 

5 

5th  of  an  8th  of  80, 

2 

5th  of  a  half  of  30, 

3 

4th  ofa  4th  of  48, 

3 

6th  of  a  3d  of  72, 

4 

3d  of  a  3d  of  36, 

4 

Exercise  25. 

Tell  what  is  the 

^ns. 

Tell  what  is  the 

AlU, 

4th  of  a  5  th  of  40, 

2 

Sd  of  an  8th  of  96, 

4 

3d  of  a  6th  of  72, 

4 

8th  ofa  3d  of  72,        ^ 

3 

oth  of  a  half  of  60, 

6 

4th  of  a  6th  of  120, 

5 

3d  of  a  6th  of  54, 

3 

5th  of  a  7th  of  105, 

S 

4th  of  a  5th  of  100, 

5 

8th  of  a  9th  of  360, 

5 

5th  of  a  6th  of  120, 

4 

3d  ofa  7th  of  63, 

3 

3d  of  an  8th  of  72, 

3 

4th  of  an  8th  of  96, 

3 

halfofa  9th  of  36, 

2 

8th  ofa  3d  of  120, 

5 

Sd  ofa  7th  of  42, 

2 

4th  ofa  4th  of  1 60, 

10 

5th  ofa  4th  of 60, 

3 

5  th  of  a  3d  of  90, 

6 

Oth  of  a  7th  of  84, 

2 

4th  of  a  7th  of  140, 

5 

ExERC 

isE  26. 

Tell  what  is 

*^ns. 

Tell  what  is 

Ans. 

2  fifths  of  25, 

10 

9  tenths  of  90, 

81 

3  fifths  of  35, 

21 

7  eighths  of  56, 

49 

4  fifths  of  75, 

60 

3  sevenths  of  42, 

18 

3  sevenths  of  21, 

9 

4  fifths  of  55, 

44 

2  ninths  of  45, 

10 

3  eighths  of  40, 

15 

6  sevenths  of  42, 

36 

3  fifths  of  60, 

36 

5  eighths  of  64, 

40 

4  sevenths  of  5^, 

32 

7  eighths  of  96, 

84 

6  sevenths  of  49, 

42 

6.  sevenths  of  35, 

30 

5  ninths  of  72, 

40 

6  sevenths  of  49, 

35 

4  fifths  of  60, 

48 

5  ninths  of  45, 

30 

7  eighths  of  64, 

5^ 

E 


50 

Exen 

lists. 

Exercise  27. 

Tell  what  is 

^ns. 

Tell  what 

is 

v^/W. 

3  times  5  times  11, 

165 

6  times  3  times  2, 

36 

4            6 

12. 

288 

8 

3 

7, 

168 

5             6 

10, 

soo 

9 

4 

5, 

180 

5             6 

9, 

270 

7 

5 

3. 

1Q5 

6             7 

8, 

336 

6 

6 

3, 

108 

6             7 

4, 

168 

2 

5 

11, 

110 

7             5 

2. 

70 

3 

6 

12. 

216 

6            6 

2. 

72 

4 

6 

10, 

240 

4            9 

8. 

288 

3 

4 

9, 

108 

5            9 

7. 

189 

5 

4 

8, 

160 

S            6 

4, 

48 

3 

5 

12, 

180 

Exercise  £8. 

Tell  what  is 

^?is. 

Tell  wha 

tis 

Ans* 

7  times  3  times 

s. 

63 

6  times  8  tim 

es7. 

336 

8            4 

64 

3 

6 

7. 

126 

6            S 

4, 

72 

4 

7 

3, 

84 

7             5 

2. 

70 

5 

9 

4, 

180 

4            5 

6, 

120 

4 

4 

4. 

64 

5             6 

7, 

210 

5 

5 

5, 

125 

6             7 

8, 

336 

6 

6 

6. 

216 

7             8 

9, 

504 

3 

4 

11. 

132 

S             8 

9, 

216 

4 

5 

12, 

240 

4            7 

8, 

224 

6 

7 

4, 

168 

5             8 

9. 

360 

5 

8 

3, 

1^0 

Exercise  29. 

Tell  what  is 

Ans, 

Tell  what 

is 

Ans, 

5  times  25, 

125 

3  times  39, 

117 

4 

26, 

104 

4 

49, 

196 

£ 

84, 

168 

5 

59, 

295 

3 

17, 

51 

6 

61. 

366 

4 

19, 

76 

7 

14, 

98 

5 

16, 

80 

S 

16. 

48 

6 

14, « 

84 

4 

18. 

72 

7 

IS, 

91 

5 

17. 

85 

S 

IS. 

120 

6 

13, 

78 

9 

21, 

189 

7 

17, 

119 

S 

29, 

58 

8 

18, 

144 

Eivercises» 

51 

Exercise  30. 

From 

take. 

^ns. 

From 

take, 

Jns 

20, 

2  thirds  of  18, 

8 

18. 

6  sevenths  of  14, 

6 

18, 

3  fourths  of  12, 

9 

29. 

8  ninths  of  27, 

5 

16, 

2  fifths  of  25, 

6 

17, 

5  sixths  of  12, 

7 

17, 

3  sevenths  of  2  8, 

5 

31, 

6  sevenths  of  21, 

15 

21, 

4  fifths  of  15, 

9 

24, 

9  tenths  of  20, 

6 

30, 

5  sixths  of  18, 

15 

27. 

3  fifths  of  25, 

12 

52, 

2  thirds  of  21, 

18 

18, 

3  sevenths  of  14, 

12 

28, 

6  sevenths  of  14, 

16 

23, 

3  eighths  of  24, 

14 

12, 

3  fourths  of  8, 

6 

26, 

4  ninths  of  27, 

14 

14, 

5  eighths  of  16, 

4 

EXERC 

19, 

ISE  3 

5  sevenths  of  21, 

4 

From 

I    take, 

^is. 

From 

take. 

^115. 

40, 

6  sevenths  of  42, 

4 

32, 

3  fifths  of  40, 

■'    8 

50, 

5  eighths  of  64, 

10 

80, 

6  sevenths  of  70, 

20 

90, 

7  eighths  of  96, 

6 

75, 

5  eighths  of  96, 

15 

36, 

6  sevenths  of  35, 

6 

39, 

8  ninths  of  36, 

7 

40, 

6  ninths  of  45, 

10 

47, 

5  sixths  of  42, 

12 

50, 

4  fifths  of  60, 

2 

26. 

3  fifths  of  40, 

2 

36, 

3  sevenths  of  42, 

18 

31, 

4  fifths  of  3  5, 

3 

26, 

9  tenths  of  20, 

8 

42, 

4  ninths  of  36, 

'^^ 

45, 

3  elevenths  of  55, 

30 

57. 

5  twelfths  of  96, 

17 

35, 

6  sevenths  of  28, 

11 

39. 

3  eighths  of  48, 

21 

J^^'iite,  The  author  of  this  work  is  of  the  opinion,  that  the  most  scientific 
aiTanj^emcnt  of  the  several  rules,  requires  Fractions  to  precede  the  Kule  of 
I'aree,  and  the  other  rules  which  usually  follow  that.  But  if  any  instructor 
slionld  think  it  most  for  the  advantag;e  of  particular  pupilc  to  attend  to  those 
rules  before  Fractions,  he  can  direct  them  to  pass  on  accordingly,  and  omit 
those  parts  of  them  in  which  fractions  are  used. 

VULGAR  FRACTIONS. 

A  fraction  is  an  expression  for  the  part  or  parts  of  a  quan- 
tity, which  quantity  is  denoted  by  unity. 

A  vulgar  fraction  is  denoted  by  two  numbers,  one  placed 
above  another,  with  a  line  between  them. 

The  number  placed  below  the  line  is  called  the  denomina- 
tor, and  that  above  it  the  numerator,  and  both  are  called 
terms. 

The  denominator  show^s  how  many  equal  parts  the  unit  or 
quantity  is  divided  into;  and  the  numerator  shows  how 
iftiany  of  those  parts  are  taken. 


2  Vulgar  Fractions, 

Thus,  if  a  pound  sterling  is  divided  into  £0  equal  parts  or 
shillings,  and  5  of  those  parts,  or  5  shillings,  are  given  to  A, 
7  to  B,  and  8  to  C,  the  share  of  each,  expressed  in  the  manner 
of  a  vulgar  fraction,  is,  A's,2^oof  a  pound ;  B'8,2^oof  a  pound ; 
and  C's,  2^0  ^^  ^  pound.* 

The  denominator  represents  the  divisor,  in  division,  and 
the  numerator  the  remainder. 

A  proper  fraction,  is  one  of  which  the  numerator  is  less 
than  the  denominator,  as  J,  or  J. 

An  improper  fraction,  is  one  of  which  the  numerator  is 
equal  to,  or  greater  than  the  denominator,  as  f ,  or  5 . 

A  single  fraction,  is  a  simple  expression  for  any  number 
of  the  parts  of  a  unit,  as  5. 

A  compound  fraction,  is  the  fraction  of  a  fraction,  as  J  of 

^"* 
A  mixed  number,  is  composed  of  a  whole  number  and  a 

fraction  together,  as  3^. 

A  whole  number  may  be  expressed  like  a  fraction,  by 
writing  1  under  it  for  a  denominator,  as^,  which  is  the  same 
as  S. 

A  common  measure  of  two  or  more  numbers,  is  that  num- 
ber which  will  divide  each  of  them  without  a  remainder. 

A  common  multiple  of  two  or  more  numbers,  is  that  num- 
ber wliich  can  be  divided  by  each  of  them  without  a  remain- 
der. 

Problem  I. 

To  find  the  greatest  common  measure  of  two  or  more 
numbers. 

Rule. 

1.  If  there  are  two  numbers  only,  divide  the  greater  by 
the  less,  then  divide  the  divisor  by  the  remainder  ;  and  so  on, 
dividing  always  the  last  divisor  by  the  last  remainder,  till 
nothing  remains ;  and  the  last  divisor  will  be  the  greatest 
common  measure  sought. 

2.  If  there  are  more  than  two  numbers,  find  the  greatest 
common  measure  of  two  of  them,  as  before  ;  then  do  the  same 
for  that  common  measure  and  another  of  the  numbers  ;  and 
so  on,  through  all  the  numbers ;  and  the  greatest  common 
measure  last  found,  will  be  the  answer. 

*  The  denominator,  instead  of  being  put  under  the  numerator,  is  some- 
times written  after  it,  separated  by  a  hyphen,  tJius,  1^2  Q7ie  half,  i3-4  t/o'ce- 
foitrthSf  10-20  ten  t-weritieths. 


I 


Vulgar  Fractions,  5S 

"j^ote.  If  it  happens  that  the  common  measure  thus  feund  is  1,  the  numbers 
are  said  to  be  incommensurable,  or  not  having  any  common  measure. 
Example. 
What  is  the  greatest  common  measure  of  336,  720,  and 
1736? 

Operation.  First,  I  take  336  and  720,  and  find  the  great- 
est common  measure  of  these,  as  follows  : 
^36)720(2  Having  divided  720  by  336,  I  find  48  re- 

67^  mainder ;  and  having  divided  the  last  divisor 

336,  by  this  remainder  48,  1  find  no  remain- 

48)336(7     der;  consequeutly  48  is  the  greatest  com- 
336         mon  measure  of  336  and  720. 

Next,  I  am  to  find  the  greatest  common 
measure  of  48  and  the  third  given  number,  1736  ;  and  I  pro- 
ceed in  the  same  way. 
48)1736(36 

144  Here,  having  divided  1736  by  48,  I  find 

8  remainder  ;  and  having  divided  48  by  this, 

I  find  no  remainder  ;  consequently  8  is  the 
greatest  common  measure  of  336,  720,  and 
1736. 
8)48(6 
48 

Problem  2. 
To  find  the  least  common  multiple  of  two  or  more  numbers* 
Rule. 

1.  Divide  by  any  number  that  will  divide  two  or  more  of 
the  given  numbers  ^vithout  a  remainder,  and  set  down  the 
quotients  and  the  numbers  not  divided  in  a  line  below. 

2.  Divide  the  second  line  in  the  same*  manner  ;  and  so  on, 
till  there  are  no  two  numbers  that  can  be  divided  without  a 
remainder. 

3.  Multiply  the  numbers  in  the  lower  line  and  the  several 
divisors  continually  together,  and  the  product  will  be  the 
least  common  multiple  required. 

E^^VMPLE* 

What  is  the  least  common  multiple  of  3,  4,  8,  and  12  ? 
Operation.  Here,  first,  I  perceive  that  4  will  divide 

4)3,     4,     8,   12  three  of  the  numbers,  to  wit,  4,  8,  and  12, 

without  a  remainder.     I  thereiore  divide 

3)3,     1,     2,     3  them  by  4,  and  set  down  their  quotients, 

'    *  1,   S,  avul  ?.    u^ii]i:r  ihp'j}  rrKiipcfivolv.  and 

1        1        o       1  -      •• 


54  Seduction  of  Vulgar  Fractions. 

I  perceive  that  3  will  divide  two  of  the  numbers,  to  wit,  S 
and  3,  without  a  remainder.  Accordingly,  1  divide  them, 
and  set  down  their  quotients,  and  the  numbers  not  divided, 
as  before  ;  and  the  third  line  is  1,  1,  2,  1,  which  being  mul- 
tiplied together,  and  by  the  divisors,  4  and  3,  gives  24,  as 
the  answer. 


REDUCTION  OF  VULGAR  FRACTIONS, 

Case  1. 
To  reduce  a  fraction  to  its  lowest  terms. 

Rule. 
Divide  both  terms  of  the  fraction  by  any  number  which 
will  divide  both  without  a  remainder,  and  those  quotients 
again  in  like  manner ;  and  so  on,  till  you  can  proceed  no 
further,  and  the  last  quotients  will  be  the  fraction  in  its  low- 
est terms. 

Or  :  Find  the  greatest  common  measure  of  the  two  terms, 
and  divide  them  both  by  it,  and  the  quotients  will  be  the  frac- 
tion in  its  lowest  terms. 

Example. 
Reduce  |~|-|-  to  its  lowest  terms. 

Operation,  Here,  I  first  divide  both  terms  by 

7)  2,  and  it  gives  |  § ,  and  these  I  divide 

2)|-||=  9^^==T4  ^^n5.  again  by  7,  and  it  gives  f-|-,  which  I 
cannot  divide  again  ;  and  consequent- 
ly |-|-  is  the  answer. 

Case  2. 
To   reduce  fractions  of  d'tferent  denominators  to  other 
fractions  of  the  same  value,  having  a  common  denominator. 

Rule. 
^lultiyly  rich  numerator  into  all  the  denominators  except 
s  own,  lor  the  new  numerators  ;  and  all  tiie  denominators 


gelher  for  a  common  denominator. 


Example. 

Reduce  J,  3,  and  |,  to  a  comn\on  denominator. 
Operat^'on.  '  Here,  I  take  1,  the  nu- 

1x3x4  =  12^   the  new  merator  of  the  first  fraction, 

?  X 2 X 4 = 1  r^  I  nu mera-  and  multiply  it  by  3 ,  and  4, 

o  X 2  X  3 =in  J  tors.  the  denominators  of  the  se- 

cond   and   third  fractions. 


2x3x4=^4  new  denomiBator.     and  it  gives  19.,  for  the  new 

numerator  of  the  first  frac- 
Tlon. 


Reductioji  of  Vulgar  Fractions.  55 

Then,  I  take  2,  the  numerator  of  .the  second  fraction,  and 
multiply  it  by  2  and  4,  the  denominators  of  the  first  and  third 
fractions,  and  it  gives  16  for  the  new  numerator  of  the  second 
fraction. 

Then  I  take  3,  the  numerator  of  the  third  fraction,  and 
multiply  it  by  2  and  3,  the  denominators  of  the  first  and  se- 
cond fractions,  and  it  gives  18  for  the  new  numerator  of  the 
third  fraction. 

Lastly,  I  take  2,  3,  and  4,  the  denominators,  and  multiply 
them  together,  and  it  gives  24  for  the  new  denominator  ;  and 
the  new  fractions  are  -^f,  ^,  and  |^. 

A''otc.  Wh«n  the  denomii^ator  of  one  fraction  is  a  multiple  cf  the  denomi- 
nator of  j\nolher,  they  may  be  reduced  to  the  sarae  denominf.tor,  by  multi- 
plying boih  the  terms  of  that  fraction  whose  denominator  is  the  smaller,  by 
such  number  as  will  make  its  denominator  equal  to  that  of  the  other. 

Example. 

Reduce  §  and-j\  to  a  common  denominator. 

Operation.  Here,  12,  the  denominator  of  one  fraction,  is  S 
times  4,  the  denominator  of  the  other.  Therefore,  f  may 
have  both  its  terms  multiplied  by  3.  Now,  S  times  3  is  9  ; 
and  3  times  4  is  12.  Therefore  -j^  is  equal  to  3,  and  its  de- 
nominator is  the  same  as  that  of  j^>  ^^'^  other  fraction.  So 
that-j^2  and  -j^  are  the  fractions,  having  the  same  denomina- 
tor. 

This  method  will  sometimes  very  much  shorten  operations 
in  addition  and  subtraction  of  vulgar  fractions. 

Case  5. 
To  reduce  a  mixed  number  to  its  equivalent  improper 
fraction. 

Rule. 
Multiply  the  whole  number  by  the  denominator  of  the 
fraction,  and  add  the  numerator  to  the  product,  and  this  will 
be  the  numerator,  under  which  write  the  denominator,  and  it 
will  be  the  improper  fraction  required. 

^       .     Example. 

Reduce  23 1  ^o  ^n  improper  fraction. 

Here,  I  first  multiply  23,  the  whole  number,  by  3,  the  de- 
nominator of  the  fraction,  and  it  gives  69,  to  which  I  add  2^ 
the  numerator  of  the  fraction,  and  it  makes  71,  for  the  new 
numerator,  under  which  1  write  3,  the  denominator  ',  and  -^ 
is  the  improper  fraction  required. 


56  EeducHon  of  Vulgar  Fr actons. 

.    Case  4. 
To  reduce  an  improper  fraction  to  its  equivalent  whole  or 
mixed  number. 

Rule. 
Divide  the  numerator  bj  the  denominator,  and  the  quotient 
will  be  the  whole  or  mixed  number  sought. 
Examples. 
Reduce  ^3*-  to  its  equivalent  mixed  number. 
5)7 1  Reduce  ~^^  to  its  equivalent  whole  number* 

—  7)56 

8  Ans. 

Case  5. 
To  reduce  a  compound  fraction  to  an  equivalent  single  one. 

Rule. 
Multiply  all  the  numerators  together  for  a  numerator,  and 
all  the  denominators  together  for  a  denominator ;  and  they 
will  form  the  single  fraction  required,  which  reduce  to  its 
lowest  terms. 

JVote.  This  operation  may  frequently  be  contracted.  When  the  same 
number  is  both  among  the  numerators  and  the  denominators,  it  may  be  struck 
out  of  both.  When  a  number  in  one  set  of  terras  will  dividet  without  a  re- 
mainder, any  number  in  the  other  set  of  terms,  the  quotient  may  be  substi- 
tuted for  the  dividend,  and  the  divisor  be  struck  out.  Or.  when  one  in 
each  set  of  terras  can  be  divided  by  the  same  number  without  a  remainder, 
the  quotients  may  be  substituted  in  their  stead. 

EXAMPE. 

Reduce  3-  of  |^  of  ^  of-|  of  y,  to  a  single  fraction* 
Operation.  I  first  multiply  all  the  numerators,  2,  3,  4,  5, 6, 
together,  for  a  new  numerator,  and  it  is  found  to  be  720; 
8nd  then  I  multiply  all  the  denominators,  3,  4,  5,  6,  7,  toge- 
ther, for  a  new  denominator,  and  it  is  found  to  be  2520. 
And  the  single  fraction  required  is  "25^%>  which  reduced  to 
its  lowest  terms,  is  y,  which  is  the  answer. 

Contraction. 
This  operation  may  be  contracted,  by  striking  out  those 
figures  which  are  the  same  in  both  sets  of  terms,  and  which, 
in  the  following  example,  are  marked  with  an  asterisk : 

•|of|of|of|off 

*  *  *  ■* 

"Where  all  the  i^i^rnf^ra^-^rs  c^re  stri^rk  fi\:t  except  the  ?,  anJ 
alithe  denc-'^*^    '  -''.k  y  ;^..\.\  :h- 


Reduction  of  Vulgar  Fractions.  57 

J\'otc.  The  reasons  for  this  contraction  are,  that  when  any  number  is  mul- 
tiplied by  another,  and  afterwards  divided  by  tlie  same,  it  is  brought  back  ?o 
what  it  was  :  and  in  a  ccwnpouad  tVaction,  the  upper  set  of  terms  are  multi- 
pliers, and  the  lower  aet  of  terms  are  divisors.  So  that,  in  the  above  ex^ 
ample,  if  you  take  2.  the  first  of  the  upper  terms,  and  multiply  it  by  3,  the 
second,  it  makes  6  ;  but  you  afterwards  have  to  divide  it  again  by  3,  whicii 
brings  it  back  to  2,  as  it  was.  Both  tlie  multiplication  by  3.  and  the  divisioa 
by  3,  may  tlierefox*e  be  omitted,  and  the  two  3's  struck  out.  And  so  of  the 
rest. 

Case  6. 
To  reduce  a  fraction  from  one  denomination  to  another. 

Rule. 
Consider  how  many  of  the  less  denomination  make  one  of 
the  greater  :  then,  if  the  reduction  is  to  a  lower  denomination^ 
multiply  the  numerator  by  that  number  ;  if  to  a  higher,  mul- 
tiply the  denominator ;  and  then  reduce  the  fraction  so  form- 
ed to  its  lowest  terms. 

Example  !• 
Reduce  -f  of  a  pound  to  the  fraction  of  a  penny. 
Operntion.  Here,  I  consider  that  240  pence  make  a  pound; 
and  as  the  reduction  is  to  a  lower  denomination,  I  multiply 
the  numerator  by  it,  and  it  is,  2  times  240,  that  is  480  which, 
is  the  numerator  of  the  fraction,  and  9  is  the  denominator : 
and  -*  9  ^,  being  reduced  to  its  lowest  terms,  is  -^-|^,  which 
is  the  answer. 

Example  2. 
»  Reduce  g  of  a  penny  to  the  fraction  of  a  pound. 
P  ^  Operation.  Here,  I  consider  that  840  pence  make  a  pound j 
and  as  the  reduction  is  to  a  higher  denomination,  1  multiply 
the  denominator  by  it,  and  it  is  6  times  240,  that  is  1440, 
i\  which  is  the  denominator  of  the  fraction,  and  5  is  the  nume- 
l  rator  -,  and-^^^^,  being  reduced,  is2"^  g,  which  is  the  answer. 

tl ,  Case  7. 

%     To  find  the  value  of  a  fraction  of  a  higher  denomination, 
I  in  whole  numbers  of  a  lower. 
f-  Rule. 

Consider  how  many  of  the  lower  denomination  make  one 
^^  of  the  higher  5  multiply  the  numerator  by  these,  and  divide 
[ji   by  the  denominator. 

Example. 
What  is  the  value  of  -|  of  a  pound  ? 

.  Operation.   Here,  1   consider  that  20  shillings  make  a 

I      pound  ;  so  1  multiply  2,  the  numerator,  by  20,  and  it  makes 

40,   v/hich  I  divide  by  3,  tiie  denominator,   and  it  gives  13 

-hillings  and  ^.    Again,  I  consider  that  12  pence  make  1 


53  Reduction  of  Vuls:ar  Fractions 


shilling  ;  so  I  multiply  1,  the  numerator  of  this  fraction,  t 
12,  and  it  makes  12,  which  1  divide  by  3,  the  denominate 
and  it  gives  4  pence.     And  the  answer  is  13s.  4d, 
Gase  8. 

To  reduce  any  given  value  or  quantity  in  a  lower  denom 
nation,  to  the  fraction  of  a  higher. 

Rlle. 

Reduce  the  given  quantity  to  the  lowest  name  in  it,  for 
numerator,  and  one  of  the  higher  denomination  to  the  san 
nam.e,  for  a  denominator,  which  reduce  to  its  lowest  terms. 
Example. 

Reduce  2  feet,  8  inches,  11  barley  corns,  to  the  fraction 
a  yard. 

Operation.  ?Iere,  I  am  first  to  reduce  this  quantity  to  t 
lowest  name  in  it,  which  is  the  fifth  of  a  barley  corn,  for 
numerator  ;  and  then  to  reduce  a  yard  to  iiflhs  of  a  barl 
corn,  for  a  denominator ;  which  is  done  as  follows  : 

ft,  in.  h.  c, 

2..8..  i^  1  yard, 

12  3 

32  inches,  3  feet, 

3  12 

97  bavlev  corns,  36  inches, 

5  "  3 

486  fifths  of  a  b.  e,  108  barley  corns, 

numerator.  5 

540  fifths  of  h.  c,  denominate 
And  the  fraction  is  y-f^,  which  being  reduced  to  its  lowe 
terms,  is  j^-,  which  is  the  answer. 

ADDITION  OF  VULGAR  FRACTIONS. 

Rule. 
Reduce  compound  fractions  to  single  ones,  mixed  nut 
bers  to  improp^.T  fractions,  these  of  different  denominatio 
to  the  same  denomination,  and  all  to  a  common  denomin 
tor;  and  the  sum  of  the  nume^-ators,  being  written  over  t 
common  denomirator,  and  that  fraction  reduced  to  its  low^ 
terms,  will  give  the  answer. 


Addition  and  bubtractton,  oj  yutgar  rractiom,        59 

Example. 
Add  -f ,  f  of  J,  and  9^,  together. 

Operation.  Here,  1  first  reduce  the  compound  fraction,  f- 
wf  J,  to  a  single  one,  and  it  makes  X5"*  I  then  reduce  the 
'mixed  number,  9^,  to  an  improper  fraction,  and  it  makes 
^^     And  the  question  stands,  ^  +  - *5  +-yV  • 

Next,  I  reduce  these  S  fractions  to  a  common  denominator, 

and  they  are  iVoV+ AVo  +  ^"W^-  '^^^J  ^^^  "«^  P^^" 
pared  for  addition,  and  the  snm  of  the  numerators  is  found 
to  be  15025,  which  being  written  over  the  common  denomi- 
nator, is  ^1  5  Q§,  and  this  reduced  to  its  lowest  terms,  is 
10  6^0'  ^vhich  is  the  answer. 

SUBTRACTION  OF  VULGAR  FRACTIONS. 
Rule. 

Prepare  the  fractions  as  in  addition,  and  subtract  the  nu- 
^  merator  of  the  less  from  that  of  the  greater ;  and  the  diffe- 
I  rence  placed  over  the  common  denominator,  and  that  frac- 
i  tipn  reduced  to  its  lowest  terms,  will  be  the  answer. 

Example. 
From  14i,  take  |of  19. 

Operation.  Here,  the  greater  number,  14^,  is  a  mixed 
number,  and  is  to  be  reduced  to  an  improper  fraction,  which 
being  done,  it  is^^.  The  less  number,  §^  of  19,  is  a  compound 
fraction,  and  is  to  be  reduced  in  a  single  fraction,  which  being 
done,  it  is  ^.  The  question  then  becomes  this,  from  ^^  take 
^.  The  next  thing  to  be  done,  is  to  reduce  these  two  frac- 
tions to  a  common  denominator,  which  being  done,  they  are 
^i^^nd  ^^y  ;  and  the  question  becomes  this,  from  ^i^tak^ 
^1^;  and  »52,  the  numerator  of  the  less,  being  taken  from 
171 ,  the  numerator  of  the  greater,  leaves  19,  which  being  pla- 
.  ijed  over  the  common  denominator,  is  |-| ,  and  this  reduced  to 
its  lowest  terms,  is  I  j^,  which  is  the  ansv/en 

MUL'tlPLICATION  OF  VULGAR  FRACTIONS. 

Rule. 
Reduce  compound  fractions  to  single  ones,  mixed  numbers 
to  improper  fractions,  and  those  of  different  denominations 
to  the  same  denomination  ;  then  multiply  the  numerators  to- 
gether for  a  new  numerator,  and  the  denominators  together 
^^>r  a  new  deaomiaator,  and  redi^ce  it  to  its  lowest  terms. 


60    Multiplicatmi  and  Division  of  Vulgar  FradlonB. 


Example. 

Multiply  12^  by  §  of  7. 

Operation,  Here,  the  first  is  a  mixed  number,  and  is  to  be 
reduced  to  an  improper  fraction,  which  being  done,  it  is  ^5*. 
The  second  number  is  a  compound  fraction,  and  is  to  be  re- 
duced to  a  single  one,  which  being  done,  it  is  7.  The  nu- 
merators t  3  and  7,  being  multiplied  together,  give  44  ^  for  a 
new  numerator  ;  and  the  denominators  5  and  3,  being  multi- 
plied together,  give  15  for  a  new  denominator  ;  and  the  pro- 
duct is  *ix">  which  being  reduced,  is  ^-f^,  or  29 f,  which  is 
the  answer. 


DIVISION  OF  VULGAR  FRACTIONS. 

Rule. 
Prepare  the  fraction  as  in  multiplication  ;  invert  the  terms 
«ftlie  divisor,  and  proceed  as  in  multiplication. 
Example. 
Divide  4  q  by  ^  of  4. 

Operation,  Here,  the  dividend  is  a  mixed  number,  and  is 
to  be  reduced  to  an  improper  fraction,  which  being  done,  it 
is  9*g^ .  The  divisor  is  a  compound  fraction,  and  is  to  be  re- 
duced to  a  single  one,  which  being  done,  it  is -^9^.  Now,  9- 
divided  by  -^9^,  is  the  same  as  ^9*  multiplied  by  y^,  the  terms 
of  the  divisor  being  inverted.  And  -9^  multiplied  by  ^,  is 
it  0»  which,  being  reduced,  is  2  2^0 ,  which  is  the  answer. 
Questions  on  the  foregoing. 

How  do  you  find  the  greatest  com- 
mon measure  of  two  numbers  f 

How  of  more  than  two  ? 

How  do  you  find  the  least  common 
multiple  ? 

How  do  you  reduce  a  fraction  to  its 
lowest  terms  ? 

How  do  you  reduce  fractions  to  a 
common  denominator  ? 

How  do  you  reduce  a  mix^d  number 
to  an  improper  fraction  ? 

How  do  you  reduce  an  improper 
fi'action  to  a  whole  or  mixed  num- 
ber ? 

How  do  you  jreduce  a  compound 
fractitm  to  a  single  one  f 

How  may  this  operatiou  be  contract- 
ed? 


What  is  a  fraction  ? 

What  is  a  vulgar  fraction  ? 

What  is  the  denominator  ? 

What  is  the  numerator  ? 

What  are  they  both  called  ? 

W^hat  does  the  denominator  show  ? 

What  does  the  numerator  aliow  ? 

Wh^t  does  each  represent ' 

What  is  a  proper  fraction  ? 

What  is  an  improper  fraction  ? 

What  is  a  single  fractian  ? 

What  is  a  compound  fraction  ? 

What  is  a  mixed  number  ? 

How  may  a  whole  number  be  ex- 
pressed like  a  fraction  ? 

What  is  meant  by  a  common  mea- 
sure ? 

Wliat  by  a  comraon  multiple  ? 


Decimal  Fractions. 


61 


How  do  you  reduce  a  fraction  from 
a  higher  denomination  to  a  lower  ? 

How  from  a  lower  to  a  higher  ? 

How  do  you  fiiid  the  value  of  a  frac- 
tion ot  a  higher  denomiuatioa  in 
whole  numbei-a  in  a  lower  ? 


riow  do  you  reduce  a  giren  value  or 

quantity  to  tlie  fraction  of  a  higher 

denomination  ? 
What  is  the  rule  for  the  addition  of 

vulgar  fractions  ?  For  subtractioa  ? 

multiplication  ?  division  ? 


DECIMAL  FRACTIONS. 

When  the  denominator  of  a  vulgar  fraction  is  1,  with  any 
number  of  cyphers  annexed,  as  10,  100,  1000,  &c.  it  is  called 
a  decimal  fraction  ;  and  instead  of  writing  the  denominator 
under  the  numerator,  the  numerator  only  is  set  down,  with  a 
point  at  the  left  hanti  of  it ;  thus,  ^^,  is  written  '5,  f-£-^  is  writ- 
ten '25,  and  yW^  is  written  'S75>  But  if  the  numerator  has 
not  as  many  places  as  there  are  cyphers  in  the  denominator, 
cyphers  must  be  prefixed  to  make  up  that  number ;  as,  xVVo> 
must  be  written  '075,  and  joWo  o  ^^^t  be  written  •001£4. 

Cyphers  at  the  right  hand  of  decimals  make  no  alteratioa 
in  their  value  ;  for,  '5,  and  '50,  and  ^oOO,  are  decimals  of  the 
same  value,  and  signify  5  tenths,  or  50  hundredths,  or  500 
thousandths.  But  if  cyphers  are  placed  on  the  left  hand  of 
the  significant  figures,  and  after  the  decimal  point,  they  de- 
crease the  value  of  those  figures  in  a  tenfold  proportion  ; 
thus,  -5  is  ^0,  but  -05  is  only  y|o,  and  -005  is  joVo* 

In  numerating  decimals,  begin  at  the  decimal  point,  and 
proceed  towards  the  right  hand.  The  first  place  is  tenths, 
the  second  hundredths,  the  third  thousandths,  the  fourth  ten 
thousandths,  the  fifth  hundred  thousandths,  the  sixth  mil- 
lionths,  and  so  on.  The  numeration  of  decimals  and  of 
whole  numbers  is  similar,  only  that  whole  numbers  proceed 
from  right  to  left,  and  decimals  proceed  from  left  to  right. 

Read,  in  words,  the  following  decimals  : 


.6 

•05 

•1 

26 

•005 

•a 

•845 

•065 

•01 

•42 

•00078 

•001 

257' 

•0708 

•0101 

•3.567 

•0^567 

•OllOl 

•467 

•00001 

•lOlOl 

•3506 

•00011 

•lion 

•9 

•00026 

•011011 

•98 

•00678 

•lOlOlOl 

•9876 

•002789 

•oioroi 

•6r 

•0102034 
F 

•011001 

^^  JDecimal  Fractions. 

Write  down  in  figures  the  following  decimals  ; 

Thirty-four  hundredths. 
Four  hundred  and  sixty-one  thousandths. 
Five  thousand  and  eleven  ten  thousandths. 
Seven  tenths. 
Eight  thousandths, 
Nine  hundredths, 
l^wenij  ei^ht  ten  thousandths. 
Sixty  eight  millionths. 
One  thousandth. 

One  hundred  and  one  millionths. 
One  thousand  one  hundred  and  one  ten  thousandths. 
Two  hundred  and  thirty -four  thousand  three  hundred  and 
five  millionths. 

One  hundred  and  one  thousand  and  one  ten  thousandths. 

One  thousand  one  hundred  and  one  millionths. 

One  ten  millionth. 

One  hundred  and  one  ten  thousandths. 

Twenty-one  ten  millionths. 

One  hundred  thousandth. 

One  millionth. 

A  mixed  number  is  made  up  of  a  whole  number  and  some 
decimal  fraction,  the  one  being  separated  from  the  other  by 
the  decimal  point,  hence  called  the  separatria;,  Ihus,  123-4, 
is  one  hundred  and  twenty- three,  and  four  tenths  ;  12-34,  is 
twelve,  and  thirty- four  hundredths  ;  1-234,  is  one,  and  two 
hundred  and  thirty-four  thousandths. 

J^'oiC'  In  stating  results,  after  operations  liave  been  performedj  it  has  been 
found  more  perspicuous  to  write  the  denominator  under,  in  the  manner  of  a 
Tad§ar  fraction. 


ADDITION  OF  DECIMALS. 

Rule. 

Place  the  numbers  so  that  the  decimal  points  shall  stand 
exactly  under  each  other,  and  then  proceed  as  in  addition  of 
whole  numbers,  putting  the  decimal  point  in  the  sum  exactly 
under  those  in  the  numbers  added. 
Example. 

Whatisthesumof4jO-f3l-47+376-004+P08+456+'7'€ 
+•05  ? 


k 


Subtraction  and  Multiplication  of  Decimals.         6S 

Operation.  Here,  I  first  set  down  the  450 ;  and  as  this  is  a 

450*  whole  number,  I   put  the  decimal  point  after  it. 

31-47  Next,  I  set  down  the  31*47,  so  that  the  decimal 

376*004  point  in  it  stands  under  the  other  decimal  point, 

1*08  and  I,  the  unit  of  the  whole  number,  stands  un- 

456*  der  0,  the  unit  of  the  first  whole  number  ;  and  4^ 

*76  the  first  decimal,  occupies  the  first  place  of  deci- 

•05  miils.     In  like  manner,  all  the  rest  are  set  down, 

—  and  added  according  to  the  rule* 


SUBTRACTION  OF  DECIMALS. 


Rule. 

Place  the  less  numlter  under  the  greater,  so  tiiat  the  deci- 
mal points  shall  be  one  Under  the  oth^r,   and  proceed  as  in 
whole  numbers,   only  putting  the  decimal  point  in  the  re- 
mainder under  those  of  the  other  numbers. 
Example. 
"What  is  the  difference  between  100-17  and  84-476  ? 
Operation* 

From  1 00- 1 7  JSTote,  As  there  is  no  figure  above  the  6, 

Take     84-476    from  which  to  subtract,  you  must  suppose  a 

cypher. 

Rem.     15*694  dns. 


MULTIPLICATION  OF  DECIMALS. 

Rule. 

Proceed  as  in  whole  numbers,  only  poi?.t  off,  in  the  pro- 
duct, as  many  dt^cimal  places  as  there  are  in  both  multiple 
cand  and  multiplier  together. 

Example. 
Multiply  -00345  by  -25. 
Operation,       Here,  I  multiply  the  significant  figures  345  by 
•00345     25,  and  get  8625  for  the  figures  of  the  product ;  but 
•25     as  there  were  five  decimal  places  in  the  multipli- 

cand,  and  two  in  the  multiplier,  there  must  be 

1725     seven  in  the  product;  so  I  prefix  three  cyphers 
690      to  make  up  the  number. 

^0008625  Ans. 


€4  Bivision  and  Reduction  of  Decimals 

DIVISION  OF  DECIMALS. 

KULE. 

Divide  as  in  whole  numbers,  and  point  ofi  as  niany  place* 
for  decimals  as  the  decimal  places  in  the  dividend  exceed 
those  in  the  divisor. 

JSf^ote.  When  there  is  a  remainder  after  division»  or  when  the  decimal 
places  of  the  dividend  are  not  so  many  as  those  of  the  divihor,  then  annex 
cyphers  to  the  dividend,  and  carry  on  the  operation  as  far  of  shall  be  thought 
requisite. 

Example. 
Divide  -0008625  by  -00345. 

Operation.  Here,   I  divide  8625,   the  significant 

'=OO345)-00O8625(-25  figures  of  the  dividend,  by  545,  the 
690  ^ns,  significant  figures  of  the  divisor, 
— —  and   get  25  for  the   figures   of  the 

1725  quotient.     Tcr  know  whether  these 

1725  are  decimals  or  not,  I  count  the  de- 

cimal places  of  the  divisor,  and  find 
them  five  ;  and  then  count  the  decimal  places  of  the  divi- 
dend, and  find  them  seven,  that  is,  two  more  than  those  of 
the  divisor.  There  must,  therefore,  be  two  decimal  places 
iu  the  quotient,  aad  I  place  the  point  before  25  accordingly. 


REDUCTION  OF  DECIMALS. 

Case  1. 
To  reduce  a  vulgar  fraction  to  its  equivalent  decimal. 

Rule. 
Divide  the  numerator  by  the  denominator,  annexing  as 
Hiany  cyphers  as  may  be  necessary,  and  the  quotient  will  h^ 
the  decimal  required. 

Example. 
Reduce  |  to  its  equivalent  decimal. 
Operation,  Here,  as  I  cannot  divide  3  by  4,  I  annex  a 

4)3-0(-75  ^ns.     cypher  to  the  3,  and  it  makes  30,  and  say,  4 
'  28  in  30,  7  times,  and  2  remains.     Again,  1  an- 

—  nex  a  cypher  to  the  2,  and  it  makes  20,  and 

20  say,  4  in  20,  5  times,  and  nothing  remains. 

20  The  quotient,  then,  is  75.  And  as  I  annexed 

two  cyphers,  or  decimal  places  to  the  3,  and 
there  were  no  decimals  in  the  divisor,  I  point  off  two  decimals 
in  the  quotient,  and  the  answer  is  -75. 


Med  action  of  Decimah.  65 

Case  2.       ^ 
To  reduce  a  quantity  consisting  of  different  denominations, 
to  its  equivalent  decimal  value. 

Rule  1. 
Write  the  given  numbers  in  order,  from  the  lowest  deno- 
mination to  the  highest,  in  a  perpendicular  column,  and  di- 
vide each  of  them  by  as  many  units  as  it  takes  of  that  deno- 
mination to  make  one  of  the  next  higher.  Set  down  the  quo- 
tient of  each  division,  as  decimal  parts,  on  the  right  hand  of 
the  dividend  next  below  it,  and  the  last  quotient  will  be  the 
decimal  required. 

Example  1. 
Reduce  l5s.  9^^^.  to  the  decimal  of  a  pound. 
Operation.  Here,  I  set  down,  in  a  perpendicu- 

4)3'  q*  lar  column,  3  farthings,  9  pence,  and 

12)9*75  d*  15   shillings.     Next,   as    4  farthings 

£0)  10-8125       s.  make   1  penny,  I  divide  the  3  by  4, 

•790625  £,  Jins.  and  it  makes  -ISd,,  which  I  set  down 
at  the  right  hand  of  90^.  the  next  di- 
vidend. Next,  as  12  pence  make  1  shilling,  I  divide  9*7 bd. 
by  12,  and  it  makes  •SlSos.,  which  I  set  down  at  the  right 
hand  of  the  15s.  Then,  as  20  shillings  make  1  pound,  I  di- 
vide 15-8 125s.  by  20,  and  it  makes  •79G625iS.,  which  i&  the 
answer. 

Example  2. 
Reduce  Zqvs,  12lb,  6oz»  14-592  drams,  to  the  decimal  of  a 
Civt 

Opernt^o7i,  Here;  I  set  down,  as  before,  14-592in 

•  fi  5  '^)  14*592  dr.        6oz,  1 2lb.  and  3  qrs.  in  a  column,  with 

^    ^  4)[3-6  i8]  a  little  space  between  them.     Next,  be- 

C  4)  6-912  oz,        cause  16  drams  make  an  ounce,  I  am 

^4)[l-7^^ffl  to   divide   \4'599.dr.  by   16  ;  but  since 

5  4)12-432  lb.         4x4  is  1  6,  if  I  divide  by  4,  and  that 

I  7)r3-lo8]  quotient  again  by  4,  it  will  be  the  same 

4)  3*444  qrs.       as  dividing  by   16,  and   will  be  more 

•861   Civt.     convenient.     So,  1  divide  14*592  by  4, 

%Bns,  and  the  quotient  is  3-648,  which  I  set 

ilown  under  the  drams,  enclosing  it  in 

brackets,  for  the  sake  of  distinction ;  and  tiien  divide  that 

quotient,  3-048,  again  by  4,  and  it  makes  •9'2o,^.,  which  I 

«et  dcwn  at  the  right  hand  of  the  %oz.    In  like  manlier,  I 

proceed  throughout }  and  the  answer  is  'BGl  €wt* 


66  Meduct'ion  of  Dechnals, 

Rule  2. 
Reduce  the  whole  quantity  to  the  lowest  denomination  of 
which  it  consists,  for  a  numerator  ;  and  reduce  one  of  that 
denomination  in  which  you  wish  your  answer  to  be,  to  the 
same,  for  a  denominator  ;  and  then  reduce  this  fraction  to  a 
decimal,  according  to  Case  1. 

Example. 
Reduce  1 5s,  6cL  to  the  decimal  of  a  £. 
Operation. 
1 5s.     6d.  l£. 

12  20 

186  d  20  s. 

numerator.  12 

240  cL  denominator. 
And  the  fraction  is  -Jff  of  a  £,,  which  is  reduced  to  a  deci- 
mal as  follows  : 

240)  18  6-0 (-775  Mswer. 
1680 
1800 
1680 


1200 
1200 
Case  S. 
To  reduce  a  decimal  fraction  to  its  value,  in  terms  of  the 
lower  denominations. 

Rule. 
Multiply  the  decimals  given,  by  as  many  units  as  it  takes  of 
the  next  lower  denomination  to  make  one  of  the  denomination 
given,  and  point  off  as  many  places  for  decimals  as  there  are 
in  the  multiplicand  ;  the  rest  will  be  whole  numbers  of  that 
lower  denomination. 

Examplf. 

Reduce  '775  of  a  £.  to  its  value. 

Gjjerat'on.     Here,  I  first  multiply  *775  by  20,  because  there 

'775       are  20s.  in  a  £,,  and  the  product  is  15*500,  the 

20     three  last  figures  being  decimals,  because  there 

were  three  decimals  in  '775,  the  multiplicand  ; 

and  the  other  two  figures,  to  wit,  i5,  are  so  many 

shillings.     Next,  I  multiply  the  decimals  of  this 

pro<^luct,  to  wit,  *500,  by    12,  because  there  are 

rf.  6-000    i2  pence  in  a  shilling ;  and  the  product  is  6-000, 


Reduction  of  Decimals. 


67 


of  which  the  6  is  6  pence  ;  and  there  being  no  more  decimals 
to  reduce  further,  the  work  is  done,  and  the  answer  is  15s.  6c?. 


i 


Questions  on  the  foregoing. 


Hiat  is  a  decima'  fraction  i* 

lu  what  manner  are  decimil  fractions 
written  ? 

What  effect  have  cyphers  placed  at 
the  right  or  left  hand  of  the  signifi- 
cant figures  of  a  decimal  ? 

In  numerating  decimals,  where  do 
yo\i  hegin  ? 

Which  way  do  you  proceed  ? 

AVhat  is  the  value  of  the  first  place  ? 
1  he  second  ?  third  ?  fourth  ?  fifth? 
sixth  ? 

"yi/'hat  difference  is  there  in  the  nu- 
meration of  whole  numbers  and  of 
decimals  .•' 

What  is  a  mixed  number,  in  decimal 


fractions  ? 

What  is  the  separatrix  ? 

In  what  manner  are  results  usually 
stated  ? 

Why  are  they  so  written  ? 

What  is  the  rule  for  the  addition  of 
decimals  p  For  subtraction  ?  mul- 
tiplication ?  division  ? 

How  do  you  reduce  a  vulgar  fraction 
to  a  decimal  f 

How  do  you  reduce  a  quantity  of 
different  denominations  to  its  de» 
cimal  value,  Ly  the  first  rule  ? 

How  by  the  second  ? 

How  do  you  reduce  a  deciraal  frae=« 
tion  to  its  value  ? 


Tell  what  is  the 

half  of  2  thirds  of  45, 
Sd  of  3  fourths  of  60, 
4th  of  4  fifths  of  60, 
5th  of  5  sixths  of  72, 
6th  of  6  sevenths  of  49, 
7th  ofr  eighths  of  56, 
8th  of  8  ninths  of  99, 
9th  of  9  tenths  of  100, 
8th  of  8  ninths  of  45, 
half  of  2  thirds  of  S6, 


T*l'1  what  Is 

6  fifths  of  55, 

7  thirds  of  24, 

8  thirds  of  12, 

9  thirds  of  27, 

1 1  fourths  of  24, 

10  sevenths  of  77, 
9  fifths  of  So, 

8  sevenths  of  49, 

9  fifths  of  60, 

9  sevenths  of  2  8, 


Exercise  32. 

.Hns. 

Tell  what  is  the 

^ns. 

15 

7th  of  7  eighths  of  48, 

6 

15 

3d  of  3  fourths  of  48, 

12 

12 

6th  of  6  sevenths  of  35, 

5 

12 

5th  of  5  sixths  of  48, 

8 

7 

4th  of  4  fifths  of  55, 

U 

7 

6th  of  6  sevenths  of  56, 

8 

11 

,  4th  of  4  fifths  of  45, 

9 

10 

Sd  of  3  fourths  of  44, 

11 

5 

5th  of  5  sixths  of4g. 

7 

12 

8th  of8  ninths  of  54, 

6 

EXERC 

isE  33. 

.^ns. 

Tell  what  Is 

An^^ 

66 

10  thirds  of  60, 

200 

56 

1 1  fourths  of  44, 

121 

32 

12  thirds  of  36, 

144 

81 

1 1  fourths  of  1 6, 

44 

66 

15  thirds  of  9, 

Ao 

110 

9  fifths  of  55, 

99 

45 

8  thirds  of  18, 

48 

56 

10  fourths  of  48, 

120 

lOS 

12  fifths  of  45, 

108 

3^ 

9  sixths  of  42, 

63 

6S 

Exercises. 

Exercise  34. 

From 

lake 

Ans. 

From 

take 

A71S, 

35  and  46, 

££, 

59 

22  and  17, 

33, 

6 

£4  and  13, 

19, 

18 

21  and  33, 

29, 

25 

33  and  12, 

£8, 

17 

37  and  l6, 

41, 

12    ' 

37  and  19, 

41, 

15 

24  and  19, 

SO, 

IS 

26  and  18, 

S3, 

11 

19  and  14, 

23, 

10 

19  and  33, 

45, 

7 

87  and  18, 

56, 

49 

£1  and  i2, 

29, 

4 

73  and  17, 

63, 

27  j 

69  and  14, 

47, 

36 

61  and  \l, 

46, 

26  i 

53  and  H, 

39, 

26 

54  and  13, 

S7, 

40  i 

44  and  14, 

49, 

9 

69  and  15, 

56, 

28  f! 

1 

Exercise  35. 

From    take 

Ans. 

From     take 

A71S*      I 

£9,     1 5  and  6  and  7, 

1 

63,     25  and  13  and  4, 

21   J 

SO,     12  and  11  and  3, 

4 

77,     1 1  and  5  and  6, 

55/ 

32,     5  and  6  and  13, 

8 

S7,     5  and  1  1  and  12, 

29  li 

45,     7  and  8  and  19, 

11 

46,     6  and  11  and  13, 

161 

43,     17  and  4  and  11, 

11 

48,     5  and  7  and  9, 

&7\ 

19,     7  and  3  and  4, 

5 

73,     11  and  12  and  6, 

44         ; 

SO,     8  and  5  and  7, 

SO 

64,     9  and  12  and  21, 

-i 

49,     8  and 

5  and  13, 

23 

25,     7  and  6  and  5, 

70,     10  and  11  and  12, 

37 

39,     11  and  13  and  5, 

10  ^ 

£5,    3  and  7  and  1 1 , 

4 

£8,     7  and  4  and  6, 

11 

EXERC 

isE  36. 

From 

take 

Ans. 

From 

take 

Ans, 

45  and  63, 

H  and  12, 

85 

27  and£l, 

1 1  and  7, 

30     , 
17    f 
3S    ' 

22  and  37, 

13  and  15, 

31 

33  and  28, 

19  and  £5, 

45  and  11, 

2£  and  13, 

21 

67  and  31, 

47  and  t3. 

77  and    8, 

S3  antl  1 6, 

36 

£2  and  13, 

19  and  12, 

4 

64  and  13, 

11  and  17, 

49 

74  and  12, 

£1  and  32, 

33 

21  and  33, 

18  and  9, 

27 

85  and  11, 

S7  and  £1, 

58 

46  and  15, 

6  and  21, 

34 

96  and  15, 

74  and  25, 

1£ 

S7  and  17, 

19  and  6, 

£9 

87  and  17, 

26  and  31, 

47 

gl  and  49, 

14  and  27, 

£9 

45  and  52, 

IS  and  49, 

35     / 

43  and  14, 

16  and  27, 

14 

57  and  51, 

24  and  19, 

45 

ExERC 

ISE  37. 

Tell  what  is  the 

Am. 

Tell  what  is  the 

Anc. 

4th  of  5  and  8  and  3, 

4 

4th  of  2  and  9  and  1, 

3 

5tb  of  7  and  4  and  9, 

4 

7th  of4and8and  2, 

£ 

7th  of  8  and  9  and  4> 

S 

4th  of  6  and  7  and  3, 

4 

Exercises. 


69 


5 


1 

2 

o 

3 
5 
£ 
5 
S 
4 
10 


Tell  what  is  the 

Sd  of  5  and  3  and  7, 
4th  of  6  and  8  and  2, 
5th  of  3  and  10  and  12, 
Sd  of  4  and  1 1  and  6, 
6th  of  9  and  2  and  7, 
7th  of  5  and  9  and  7, 
3d  of  1  and  '8  and  9, 

Tell  what  is  the 

8th  of  a  12th  of  96, 
10th  of  a  12th  of  240, 
8th  of  an  8th  of  128, 
6th  of  a  6th  of  108, 
5th  of  a  5th  of  125, 
12th  of  a  12th  of  288, 
4th  of  a  20th  of  400, 
6thofan8thof  144, 
5th  of  a  l6thof  320, 
3d  of  a  I5thof450, 

Tell  what  is  the 
half  of  2  thirds  of  SO, 
4thof  4fifthsof  100, 
3d  of  2  thirds  of  72, 
half  of  2  fifths  of  40, 
3dof3  fifths  of  45, 
half  of  3  fourths  of  40, 
4th  of  3  fourths  of  16, 
4th  of  3  fourths  of  48, 
Gthof2  thirds  of  36, 
4th  of  3  fifths  of  80, 

Tell  what  is 

2  thirds  of  3  fourths  of  48,  24 

12 


Tell  what  is  the 
9th  of  3  and  1 1  and  4, 
8th  of  5  and  7  and  12, 
6th  of  8  and  11  and  5, 
4th  of  6  and  7  and  1 1, 
4th  of  7  and  5  and  8, 
6th  of  7  and  3  and  2, 
6th  of  3  and  1 1  and  4, 
Exercise  38. 

.dtis    I  Tell  what  is  the 

4th  of  a  12th  of  240, 
5th  of  a  10th  of  500, 
5th  of  an  18th  of  360, 
4th  of  a  l6thof  320, 
3d  of  a  10th  of  330, 
5th  of  a  15th  of  300, 
4th  of  a  20th  of  800, 
8th  of  a  10th  of  400, 
3d  of  a  9th  of  270, 
3d  of  an  8th  of  240, 


Exercise  39. 


Tell  what  is  th€ 

3d  of  9.  thirds  of  45, 
4th  of3  fourths  of  96, 
5th  ofS  fifths  of  100, 
7th  of  5  sevenths  of  49, 
12thof  7eighthsof  96, 
4th  of  5  eighths  of  64, 
3d  of  4  fifths  of  75, 
7thof  3fifdisof35, 
4th  of  3  fifths  of  60, 
8th  of  4  sevenths  of  56, 
Exercise  40. 
^ns.  Tell  what  is 

3  fifths  of  5  sixths  of  54, 


Ans, 

10 

20 

16 

8 

9 

15 

S 

9 

4 

12 


Ans, 
2 
3 
4 
6 
5 
2 


Ans. 

5 
10 

4 

5 
11 

4 
10 

5 
10 
10 

Ans. 
10 
18 
12 
5 
7 
10 
20 

s 

9 

4 

AH9. 

27 


3  fourths  of  4  fifths  of  20, 

S  fifths  of  5  sixths  of  30,  15 

2  thirds  of  3  fifths  of  45,  18 

3  fourths  of  4  fifths  of  45,  27 
2  thirds  of  3  fourths  of  24,  12 

2  thirds  of  3  fourths  of  36,  18 

4  fifths  of  5  sixths  of  48,  32 

5  sixths  of  6  sevenths  of  28,  20 

3  fourths  of  4  fifths  of  35,  21 


2  fifths  of  5  sevenths  of  42, 12 

2  thirds  of  3  severxths  of  49, 14 

3  fourths  of  4  fifths  of  60,  36 
3  fifths  of  5  sevenths  of  35,  15 

2  sevenths  of7  ninths  of  72,  16 

3  sevenths  of  7  eighths  of  56,21 
2  ninths  of  9  elevenths  of  99,18 
5  ninths  of  9  twelfths  of  CO,  25 
5  sevenths  of  7  eighths  of  32^24 


Exercises. 


Tell  what  is  the 
6th  of  an  8th  of  12  tunes  12, 
4th  of  a  20th  of  5  times  80, 
6th  of  a  6th  of  9  times  12, 
10th  of  a  12th  of  6  times  40, 
5th  of  a  16th  of  8  times  40, 
3d  of  a  15th  of  9  times  50, 
5th  of  an  18th  of  8  times  45, 
4th  of  a  16th  of  8  times  40,. 
5th  of  a  15th  of  5  times  60, 
4th  of  a  10th  of  6  times  40, 


Exercise  41. 


AnsATfAl  what  is  the 

3  3ii  of  an  8th  of  4  times  60, 
5  3d  of  a  7th  of  6  times  70, 

3  8tfi  of  a  3(i  of  6  tunes  12, 
2  5th  ora4th  of  5  times  12, 

4  5th  of  a  6tli  of  10  times  12, 
10  3d  of  a  6th  of  12  tiy:ies  9, 

4  51h  of  a  7th  of  5  times  21, 

5  8th  of  a  9th  of  9  times  40, 

4 1 5th  of  a  6th  of  V2  times  20, 

5l3dofau8th  of  6  times  20, 


Tell  what  is  the 
4th  ofa  5th  of  27  and  13, 
3dof  a6'hof  15aftd57, 
5th  ofahalfof39  and  21, 
3d  ofa6thof  19aMd35, 
4th  ofa  5th  of  87  and  13, 
5th  of  a  6th  of  65  and  55, 
3d  of  an  8th  of  39  and  33, 
half  ofa  9th  of  25  and  M, 
3d  ofa  7th  of  19  and  23, 
5th  ofa  4th  of  46  and  14, 

Tell  what  is  the 
3d  of  13  and  a  3d  of  24, 
3d  of27  and  a  4th  of  ^8, 
5th  of  35  and  a  6th  of  48, 
4th  of  24  and  a  3d  of  33, 
8th  of  40  and  a  5th  of  45, 
3d  of  36  and  a  7th  of  35, 
half  of  1 8  and  an  8th  of  32, 
5th  of  35  and  a  9th  of  45, 
7th  of  49  and  a  6th  of  48, 
8th  of  56  and  a  3d  of  45, 


Exercise  42. 


14 
16 
15 
17 
14 
17 
13 


From  the 
half  of  26, 
3ddf27, 
3d  of  33, 
4th  of  48, 
5th  of  45, 
6th  of  42, 
7th  of  28, 
half  of  50, 
3d  of  66, 
4th  of  44, 


take  the 
8th  of  3^, 
9th  of  27, 
4th  of  32, 
eth  of  66, 
9th  of  36, 
8th  of  24, 
8th  of  J  6, 
3d  of  66, 
6th  of  48, 
5th  of  40, 


Ans.  Tel!  what  is  the 

2  6th  of  a  7th  of  38  and  46, 
3d  ofan8thofl9and  77, 
8th  ofa  3d  of  27  and  45, 
4th  ofa  6th  of  87  and  33, 
5th  of  a  7th  of  63  and  42, 
8th  of  a  9th  of  195  and  165, 
6thofan8thof97  and  47, 
5thofa5thof  83and42, 
7th  of  an  8ih  of  84  and  28, 
4ihofa5lhof53  and  27, 

Exercise  4:^. 

Tell  what  is  the 
lIthof44  and  a  half  of  56, 
9th  of  63  and  a  3<1  of  96, 
halfof64anda3dof27, 
3d  of21  and  a  3d  of  99, 
5th  of  60  and  an  8th  of  96, 
9th  of  72  and  a  2d  of  72, 
„  8th  of  160  and  a  3d  of  120, 
12  9th  of  270  and  a  5th  of  250, 
15J8th  of  320  and  a  7th  of  140, 
22  half  of  150  and  a  3d  of  45, 

Exercise  44. 


Ans. 
9 
6 
3 
1 
5 
4 
2 
3 
14 
3 


From 
3d  of  63, 
4th  of  32, 
3d  of  27, 
half  of  48, 
3u  of  69, 
4th  of  88, 
5th  of  60, 
6th  of  54, 
7th  of  84, 
8th  of  96, 


take 

6lh  of  54, 
7  th  of  35, 
8th  of  61, 
5th  of  100, 
4th  of  72, 
5th  of  45, 
7th  of  35, 
8th  of  48, 
9th  of  81, 
12th  of  96. 


i^- 


Exercises, 

7i 

EXER 

ciSE  45. 

Tell  wliat 

multiplied 

Tell  what 

multiplied 

is  the 

bj  the 

Jns 

is  the 

by  the 

Jim. 

Balfofie 

3d  of  21, 

53 

8th  of  48, 

7th  of  56, 

48 

4th  of  20, 

half  of  12, 

30 

4lhof36, 

6th  of  42, 

63 

6th  of  42, 

3dof», 

21 

3d  of  33, 

9th  of  54, 

66 

nu  of  28, 

5th  of  30, 

24 

half  of  18, 

8th  of  48, 

54 

8th  of  40, 

7th  of  35, 

25 

5th  of  J5, 

half  of  24, 

84 

9th  of  45, 

3d  of  33, 

55 

8th  of  32, 

3d  of  33, 

44 

10th  of  80 

9th  of  27, 

24 

7th  of  21, 

4th  of  20, 

15 

12th  of  36 

4th  of  S6, 

27 

12th  of  108,       7tiJof21, 

27 

6th  of  54, 

8th  of  56, 

63 

3d  of  36, 

5th  of  45, 

108 

7th  of  42, 

5th  of  35, 

42 

9t|iof63, 
ISE  /l6. 

6th  of  42, 

49 

PjXERc 

Tell  what 

divided 

TcU  w!>.at 

divided 

is  the 

by  the 

Ans. 

is  the 

by  the 

J71S, 

half  of  24, 

8th  of  16, 

6 

8th  of  160, 

9th  of  45, 

4 

3d  of  36, 

4th  of  24, 

2 

5tbof200, 

6tii  of  30, 

8 

5th  of  60, 

9th  of  27, 

4  - 

4th  of  120, 

7th  of  35, 

6 

4th  of  72, 

8th  of  24, 

6 

9th  of  108, 

7th  ol  28, 

3 

6th  of  48, 

9th  of  18, 

4 

3d  of  240, 

12th  of  96, 

10 

7th  of  63, 

7th  of  21, 

3 

6th  of  96, 

11th  of  22, 

8 

7th  of  56, 

12th  of  24, 

4 

7th  of  84, 

7th  of  21, 

4 

8th  of  96, 

9th  of  36, 

3 

9th  of 81, 

6th  of  18, 

3 

9th  of  81, 

3d  of  9, 

6th  of  72, 

9th  of  27, 

4 

6th  of  90, 

8th  of  24, 

5 

8th  ©f  112, 

llth  of  22, 

7 

EXERC 

ISE  47. 

Tell  the  least  coramon  multip 

eof 

Tell  the  least  common  mulupl 

eof 

2,    3, 

and    4,     Ans 

.  12 

3,    6, 

and    8,     Ans 

24 

8,    3, 

and    2, 

24 

2,    5, 

and    6, 

30 

2,    4, 

and    6, 

12 

4,    6, 

and    9, 

36 

6,    4, 

and  10, 

60 

8,  10, 

and    4, 

40 

12,    9, 

and    6, 

36 

S,    6, 

and    9, 

18 

2,    3, 

and    9, 

18 

4,    8, 

and    6, 

24 

6,  10, 

and  20, 

60 

2,    3, 

and    6, 

12 

3,    9,  and  15, 

45 

3,    4, 

and    8, 

24 

2,    5, 

and  10, 

20 

4,    6, 

and    3, 

12 

3,    5, 

and    6, 

SO 

6.    7. 

and    3, 

42 

PROPORTION, 

Is  the  relation  which  one  qiiantitj  has  to  another. 

Arithmetical  Proportion  will  be  treated  of  hereafter. 

Geometrical  Proportion  is  that  relation  of  two  quantities 
•of  the  same  kind,  which  arises  from  considering  what  part 
the  one  is  of  the  other,  or  liow  often  one  is  contained  in  the 
#ther. 


72  Froportion. 

Four  quantities  are  said  to  be  proportional,  when  the  first 
is  the  same  part  or  multiple  of  the  second,  that  the  third  is 
of  the  fourth.  Thus,  2,  4,  3,  6,  are  proportional,  because  the 
first  is  one  half  the  second,  and  the  third  is  one  half  the 
fourth.  To  denote  this  proportion,  they  are  written  thus  : — 
2  :  4  :  :  3  :  6  ;  that  is,  as  2  is  to  4,  so  is  3  to  6. 

Direct  proportion  is  when  one  quantity  increases  in  the 
same  proportion  as  another  increases. 

Inverse  proportion  is  when  one  quantity  increases  in  the 
same  proportion  as  another  diminishes. 

If  four  quantities  be  in  geometrical  proportion,  the  product 
of  the  two  means  will  be  equal  to  the  product  of  the  two  ex- 
tiem.es.     Hence, 

If  the  product  of  the  two  means  be  divided  by  either  ex- 
treme, the  quotient  will  be  the  other  extreme ;  which  is  the 
foundation  of  the  following  rule. 

SIMPLE  PROPORTION,  OR  THE  RULE  OF  THREE, 

Is  a  composition  of  multiplication  and  division,  which 
teaches  how  to  find  the  fourth  term  of  a  proportion  from  three 
that  are  given. 

1 .  Observe  that  two  of  the  given  numbers  are  of  the  same 
name,  but  one  greater  than  the  other. 

2.  Observe  that  tlie  other  given  number  is  of  the  same 
name  with  the  number  sought. 

Rule. 

1.  Set  down  for  your  second  or  middle  term,  that  number 
which  is  of  the  same  name  with  the  answer,  or  number  sought. 

2.  To  know  how  to  place  the  other  two  numbers,  consider 
whether  the  answer  should  be  greater  or  Less  than  the  middle 
term.     If  greater,  put  the  greater  number  at  the  right  hand,  , 
for  the  third  term  ;  if  less,  the  less  ;  and  put  the  other  num- 
ber on  the  left,  for  the  first  term. 

3.  Multiply  the  second  and  third  terms  together,  and  di- 
vide their  product  by  the  first ;  the  quotient  will  be  the  an- 
swer. 

JVote.  If  the  first  and  third  terms  consist  of  different  denominations,  re- 
duce them  both  to  the  same  ;  and  if  the  middle  term  is  a  compound  number, 
reduce  it  to  the  lowest  denomination  mentioned  ;  then  the  answer  will  be  of 
the  same  denomination.  If,  after  division,  Uiere  is  any  remainder,  and  it 
can  be  reduced  to  a  lower  denomination,  reduce  it,  and  divide  again,  and  so 
on.  And  if  the  answer,  when  found,  can  be  reduced  to  a  higher  denomina- 
tion, reduce  it  accordingly. 

Example  1. 
If  7  Cwt  1  qvn  of  sugar  cost  £'ZQ  •.  10 ..  4,  what  will  43  Cwt* 
2qr,  cost? 


Froportion.  73 

Here,  I  first  consider  of  what  name  the  answer  will  be, 
and  conclude  it  will  be  pounds,  shillings  and  pence.  So  I  set 
down  £^6  ..  10  ..  4,  for  the  second  or  middle  term.  Then, 
to  determine  where  to  place  the  other  two  terms,  I  consider 
whether  the  answer  will  be  more  or  less  than  the  middle 
term.  I  conclude  it  must  be  more,  because  43  Cwt.  will  cost 
more  than  7  Cwt  So  I  put  43  CwU  ^2qr,  on  the  right  hand 
for  my  third  term,  and  7  CwU  \qr.  on  the  left,  for  my  first, 
thus  : 

First.  Second.  Third. 

Cwt,  qr.        £      s.     d,  Cwt  qr. 

7..  1     :     26..  10..  4     :  :     43  ..  2 

The  next  thing  is  to  multiply  the  second  and  third  terms. 

But,  as  they  consist  of  different  denominations,  they  must 

first  be  reduced  to  the  lowest  mentioned,  and  the  first  term 

must  also  be  reduced  to  the  same  denomination  with  the  third, 

This  is  done  as  follows  : 

CwU  qr.  =g      s.     d.  Cwt  qr, 

7..  I  g6.,  10..  4  43..  2 

4  20  4 

S9  qrs.  530  s.  174  qrs, 

12 
6364  d. 
And  the  question  stands, 

qrs.  d.  qrs^ 

29     :     6364     :  :     174     : 
I  have  now  to  multiply  the  second  and  third  terms  toge- 
ther, and  divide  by  the  first,  and  the  quotient  will  be  the  an- 
swer in  pence. 

29 


:     6364     ::     174 

174 

25456 
44548 
6364 

The  quotient  is  38184 
pence,  which  being  re- 
duced to  pounds,  gives 
^159  ..2,    for  the  an- 
swer. 

1107336(38184] 
87 

237 

24$ 

232 

232 

5^ 

116 

29 

116 

24 

G 

74  Froportion* 

Example  2. 
If  48  men  can  perform  a  piece  of  work  in  24  days,  how 
many  men  can  do  it  in  192  days  ? 

days.     men.  days. 

192     :     48     :  :     24     : 
.24 


192 
96 


192);  152(6  mc»,  Ajis. 
1152 
5.  If  ^100  in  12  months  gain  a  certain  interest,  what  sum 
will  gain  the  same  in  8  months  ?  ^ns.  ^150. 

4.  If  S  ounces  of  silver  cost  IT'S.,  what  cost  48  oz.  ? 

Ans.£\^..12. 

Questions  on  the  foregoing. 


iWbat  is  Proportion  ? 

What  is  Geometrical  Proportion  ? 

When   are  four  quantities  propor- 
tional ? 

How  is  that  proportion  denoted  ? 

What  is  direct  proportion  ? 

What  is  inverse  proportion  ? 

What  is  the  foundation  of  the  Rule 
of  Three? 

What  is  th«  Rule  of  Three  ? 

How  many  numbers  are  given  ? 

What  is  the  first  observation  respect- 
ing the  given  numbers  ? 

What  is  the  second  ? 

Which  number  do  you  set  down  fir?  ^ 
and  for  what  ? 

How  do  you  find  out  how  to  set  down 
ihe  others  I 


When  do  you  put  the  greater  at  the 
right  handr  and  when  the  les«  ? 

When  you  have  your  question  stated, 
which  numbers  do  you  multiply 
together  ? 

By  which  do  you  divide  ; 

When  your  first  and  third  terms 
consist  of  different  denominations, 
what  is  to  be  done  ? 

What,  when  your  middle  term  con- 
sists of  different  denominations  ? 

Of  what  denomination  will  the  quo- 
tient be  ? 

If  there  is  any  remainder  after  divi- 
sion, what  is  to  be  done  ? 

If  the  quotient  is  in  a  lower  deno- 
mination, what  is  to  be  done  ? 


COMPOUND  PROPORTION, 

OR  THE  DOUBLE  RULE  OF  THREE, 

Is  that  which  embraces  such  questions  as  require  two  or 
more  statings  in  the  Single  Rule  of  Three. 

The  number  of  terms  given  will  always  be  odd,  that  is, 
there  will  be  five,  or  seven,  or  nine,  &c. 

If  more  than  five  terms  be  given,  work  by  several  statings 
in  the  Single  Rule  of  Three. 

If  five  numbers  be  given  to  find  a  sixth,  three  of  them  will 
be  a  supposition,  and  tw©  a  demand.  Thea  work  by  the  fal- 
lowing 


Froportioru  T5 

Rule. 

!.  Write  down  the  three  terms  of  supposition,  so  that  the 
principal  cause  of  gain  or  loss  shall  possess  the  first  place  ; 
the  meansy  that  is,  the  distance,  time,  &c.  the  second  ;  and 
the  effect  produced,  that  is,  the  gain, loss,  &c.  the  third  ;  then 
place  the  other  two  numbers  under  those  of  the  same  name. 

2.  If  the  blank  place  falls  under  the  third  term,  multiply 
the  two  first  together  for  a  divisor,  and  the  rest  for  a  dividend. 

S.  But  if  the  blank  falls  under  any  other  term,  multiply 
the  third  and  fourth  terms  together  for  a  divisor,  and  the  rest 
for  a  dividend.  And  the  quotient  will  be  the  answer,  or 
term  sought. 

JYote.  If  any  of  the  terms  consist  of  different  denominations,^  they  must  lie 
reduced,  as  in  the  single  rule  of  three. 

Example  1. 

If  3  men,  in  4  days,  eat  6lb»  of  bread,  how  much  will  suf- 
fice 6  men  for  12  days  ? 

Here,  to  know  how  to  state  the  question,  I  consider  what 
are  the  terms  of  supposition,  and  find  them  to  be,  that  3  men, 
in  4  days,  eat  6lb.  of  bread.  These,  then,  are  to  be  the  three 
first  terms.  Next,  to  kaow  in  what  order  to  place  them,  I 
consider  which  is  the  operating  cause,  and  find  it  to  be  3  men. 
This,  then,  is  the  first  term.  I  then  consider  which  is  the 
means,  or  time,  and  find  it  to  be  4  days  ;  therefore  this  is  the 
second  term.  The  6lb*  of  bread  is  the  effect,  raid  is  therefore 
the  third  term.  And  they  stand  thus : 
men.  days,  lb. 
3  4  6 

I  then  wish  to  know  where  to  place  the  other  two  ;  and  the 
rule  is,  under  those  of  the  same  name.  And  the  whole  stands 
thus  : 

men.      days.         lb. 

3  4  6 

6  12 

Now,  to  know  which  terms  to  multiply  for  a  divisor,  and 
which  for  a  dividend,  I  consider  where  the  blank  falls ;  and 
it  being  under  the  third  term,  I  multiply  the  first  and  second 
for  a  divisor,  and  the  rest  for  a  dividend. 


T€  Compound  Proportion, 


I)ivisor, 
3 

4 

Dividend* 
6 
6 

12 

36 
12 

12)432(36  Quotient 
36 

72    And  the  answer  is  56lh 
72 
Example  2. 
If  7  men  ean  reap  84  acres  of  wheat  in  12  clays,  how  manj 
men  can  reap  100  acres  in  5  days  B 
Stated  thus: 

men,        days,        acres* 
7  12  84 

5  100 

Here,  because  the  blank  falls  not  under  the  third  term,  I 
multiply  the  third  and  fourth  terms  together  for  a  divisor,, 
and  the  rest  for  a  dividend. 

Divisor,  Dividend. 

84  7 

5  12 

420  84 

100 

4£0)8400(20  Quotient. 

840     And  the  answer  is  20  men* 

Contractions. 
Questions  in  this  rule  may  be  contracted  several  ways. 
1.  When  the  same  number  is  found^in  both  divisor  and  divi- 
dend, strike  it  out  of  both.  2.  If  a  number  in  one  will  divide 
a  number  in  the  other  without  a  remainder,  strike  out  both, 
and  put  the  quotient  of  the  one  divided  in  its  place.  3.  If 
both  divisor  and  dividend  can  be  divided  by  the  same  number 
without*  a  remainder,  their  quotients  may  be  substituted.— 
The  reason  for  which  contractions  is,  that  when  a  nuriaber  is 
multiplied  by  any  figure,  and  the  product  divided  by  the  same 
figure,  the  number  remains  just  as  it  wa3  at  first. 


Compound  Proportion*  77 

Take  the  preceding  examples. 

men,         days,  lb. 

*3  *4  6  6 

6         m  6 

36  Ans, 
Here,  3x4  =  12.  This  being  the  divisor,  and  fliere  being 
\1  also  in  the  dividend,  the  3,  4,  and  12,  marked  with  an  as- 
terisk, may  be  struck  out;  and  there  remains  only  6  and  6, 
parts  of  the  dividend,  to  be  multiplied  together,  which  makes 
36  ;  and  there  being  none  of  the  divisor  left,  36  is  the  answer. 
In  the  second  example, 

men,        days,        acres, 
'^r  n2  ^84  5)100 

5  100  ^^    ' 

20  Ans. 
84  and  5  form  the  divisor,  and  7,  12  and  1 00  the  dividend , 
rxl2  is  84,  which  being  struck  out  of  the  dividend,  and  84 
being  struck  out  of  the  divisor,  there  remains  100  to  be  di- 
vided by  5,  which  gives  20  for  the  answer. 

This  method  of  contraction  may  be  used  also  in  the  single 
rule  of  three,  and  in  division,  and  in  any  other  operations  of 
a  similar  nature. 


Questions  on  the  foregoing, 

What  is  the  Double  Rule  of  Three  ? 

If  more  than  five  nunibers  Skve  gives, 
how  must  you  proceed  ? 

If  five  arfe  given,  what  will  they  be  ? 

How  ilo  you  slate  questions  in  this 
rule  ? 

How  do  yoa  proceed  after  your  ques- 
tion is  stated } 


What  if  any  of  the  numbers  consi  t 

of  difTcrcnt  denomiriutions  ? 
Ho  ;*  may  questionsr  in  this  rule  be 

f  onlracted  ? 
On  what  reason  are  such  contractioii. 

founded  I 
Can  such  contractions  be  made  in  any 

other  rule  ? 


THE  SINGLE  RULE  OF  THREE; 
IIS  VULGAR  FRACTIONS. 

Rule. 

1.  State  the  question  as  in  the  Rule  of  Three  in  vvhok. 
numbers. 

2.  Prepare  the  fractions  as  in  multiplication   of  vulgar 
fractions. 

3.  Invert  the  terms  of  the  divisor,  and  proceed  as  in  mul- 
tiplication. 

A'^ote,  Th^  operation  ^lay  frequently  be  contracted,  as  in  Case  5  of  Rc'  * 
i^ction  of  vSear  Fractions. 


78  Single  Rule 'of  Tliree  in  Vulgar  Fractions. 

Example. 

If  f  of  a  yard  cost  /^  of  a  £.,  what  cost  y\  ol  a  yard  ? 

Operation.  Here,  in  order  to  state  the  question,  I  consider 
whether  the  answer  needs  to  Ije  more  or  less  ;  and  finding  it 
must  be  less,  I  say, 

yd,         £.  yd, 

f     *     T T     •  •     t\     •    i^^^  Answer. 

Next,  I  consider  whether  the  fractions  need  any  reduction, 
to  prepare  them  for  the  further  operations,  and  find  that  they 
do  not. 

Then,  I  invert  the  terms  of  that  which  will  be  the  divisor, 
|,  and  it  makes  f  ;  and  multiply  them  all  together,  as  follows: 

which  being  reduced^  is  |^  of  a  £.  or  3s.  Ad.,  which  is  the  an- 
swer. 

Contraction. 
This  operation  may  be  contracted  as  follows  : 

*     *       * 

3  '^  1  5  ^  1  4  b  • 

*  #  *r 

3  2 
Here,  there  being  a  3  in  the  line  of  numerators,  and  ano- 
ther in  the  line  of  denominators,  they  are  both  struck  out. 
Next,  the  denominator  15  is  divided  by  the  numerator  5,  and 
the  quotient  3  substituted  for  the  15,  and  both  1 5  and  5  are 
struck  out^  Then,  the  denominator  14  is  divided  by  the  nu- 
merator 7,  and  the  quotient  2  is  substituted  for  the  14,  and 
the  14  and  7  are  struck  out.  Now,  all  the  numerators  being 
struck  out,  I  put  down  1  for  a  numerator ;  and  there  being 
nothing  left  of  the  denominators  but  3  and  2,  I  multiply  them 
together,  and  it  makes  6,  and  the  ansv/er  is  },  as  before. 


THE  DOUBLE  RULE  OF  THREE 
IN  VULGAR  FRACTIONS. 

Rule. 

1.  State  the  question  as  in  the  double  rule  of  three  ia 
whole  numbers. 

9.  Prepare  the  fractions  as  in  multiplication  of  vulgar 
fractions. 

3.  Invert  the  terms  of  those  w^hich  will  form  the  divisor, 
and  proceed  as  in  multiplication. 


Double  Ruh  of  Three  in  Vulgar  Fractions.  75 

Example. 

When  12  persons  use  llib.  of  tea  per  naontli,  how  much 
, should  a  famii J  of  8  persons  providefor  ^  a  year  ? 

Operation,  Here,  I  first  state  the  question  according  to  the 
rule  in  whole  numbers  : 

2:iersons»        yr.  lb. 

12  tV  H 

8  I 

Next,  T  prepare  the  fractions  as  in  multiplication,  and  it 
becomes  : 

persons.^        yr.  lb. 

1  12  8  . 

AX 

The  blank  falling  und^r  the  third  term,  I  conclude  the  first 
and  second  will  form  the  divisor  ;  and  these  being  inverted, 
all  are  multiplied  together,  as  follo^vs  : 

1  2  '^    1     '^  8  '^  1  '^  2  1  9  2'* 

The  product,  being  reduced,  is  f,  or  4^,  which  is  the  answer. 
This  may  be  contracted  as  follows  : 


•THE  SINGLE  RULE  OF  THREE,  AND  THE 
DOUBLE  RULE  OF  THREE,  IN  DECIMALS. 

Questions  in  these  rules  are  stated  and  performed  as  m 
whole  numbers,  regard  being  had  to  the  placing  of  the  deci- 
mal point,  according  to  the  rules  for  the  multiplication  and 
division  of  decimals* 

Example  1. 
If  3*5?&,  of  tea  cost  U^.  what  cost  5-25i^.? 
Operation, 
lb.  £.  lb. 

3-5     :     1-4     :  :     5-25     : 
1-4 


5'5)7-35G(2-10^,  Ms. 
70 

35 


S9 


The  Rule  of  Three  in  Decimals. 


Example  2. 

If  4*3  bushels 

of  wheat  serve  1-6  men  2-5  months. 

how 

inj  bushels  will  serve  5  men  3-2  months  ? 

Operation. 

men.    months. 

5ms/?. 

dividend. 

1-6        2-5 

4-2 

3-2 

5            S'2 

4-2 
divisor.  — > 

1-6  128 

150  13-44 

m         5 


4-00)67-20^^l6-8 
400 


*^ns. 


27  m- 

2400 


Mow 


3200 

3200 

Questions  on  the  forxgoing. 

How  do  you  perform  questions  in  the 
Double  Rule  of  Three  in  Vulgar" 


do  3'OU  state  questions  in  the 
Single   Rule  of  Three  in  V^ulgar 


Fractions  ? 
How  do  you  prepare  the  fractions  ? 
How  do  }  ou  proceed  then  ? 
How  can  the  operation  be  contracted? 

Exercise  48, 


Flections  f 
How  do  you  perform  questions  in  tlie 
Single  and  Double  Rule  of  Three 
Decimals  ? 


[f  ^Ih.  cost  5s.  what  cost  9/5.  ?            Ms.  15s. 

5     .     .     4    . 

.     .      15     .     .     . 

.        12 

7    .     .     9     . 

.     .     35     .     .     . 

.        45 

6     .     .     8     . 

.     .     24     .     .     . 

.       32 

7     .     .     8     . 

.     ,     21     .     .     . 

.       24 

5     .     .     9     . 

.     .     20     .     .     . 

.       36 

9     .     .     6     .     , 

.-    45     .     .     . 

.       30 

5     .     .     8     ,     , 

.     \5     .     .     .     , 

2i 

6     .     .     4     .     . 

.     24     .     .     .     , 

16 

7     .    .     6     .     . 

.     21     .     .     .     . 

18 

8     .     .  10     .     , 

.     40     .     .     .     . 

50 

9     .     .     4     .     . 

.     27     .     .     .     . 

12 

11     .     .  12     .     . 

.     55     .     .     .     . 

60 

7     .     .     3     .     , 

•     42     ...     . 

18 

4     .     .     9     .     . 

.     16     .     .     .     . 

56 

K                       (\ 

<?<;                 - 

AO. 

Fractice* 


81 


Exercise  49. 
If  6  cost  4s.  what  cost  9  ? 


8 
3 
3 

3 
3 
3 
3 

4 
4 

2 

4 
4 
2 


6 
5 
4 

7 
8 
2 
4 
5 
5 
3 
5 
7 
7 
3 
9 


6 
2 
5 
8 
2 
5 
7 
5 
5 
7 
5 
3 
5 
9 
3 


s. 
Ans.  6 
"       4 

-  3, 
.       6 

-  18 

-  5, 
3, 

-  9 

-  8 

-  6 

-  5 

-  12 

-  10 

-  8 

-  6 

-  13 


.0 
.6 
.4 
.8 
.8 
.4 
.4 
.4 
4 
..3 
.3 
..6 
..6 
.9 
..9 
..6 


PRACTICE, 

Is  a  short  method  of  finding  the  value  of  any  number  of 
articles,  from  the  price  ©f  one  being  known. 

Table  of  PROPORfiONAL  Parts. 

MONEY.  WEIGHT. 

d.        £.  lb.       CwU 

.  0  is  ^V  7  is  ^V 

.8  is  -,\  8  is  j\ 

.  0  is  yV  14  is  i 

-  6  is  \  16  is  I 

.4  is  1  28  is  J- 

..  0  is  i  56  is  I 
,.  0  is  i 
..  8  is  i 
..  0  is  i 

Case  1. 
When  the  quantity  is  of  one  denomination,  and  the  price 
ftf  one  is  also  of  one  denomination. 

Rule. 
Multiply  the  quantity  by  the  price,   and  the  product  will 
be  the  answer,  in  the  same  denomination  with  tlie  price. 


q.      d. 

1  isi 

2  is  i 

s. 
1 
1 

o 

d.         s. 

2 

1     is  -,\ 

3 

U  is  i 
2     is  i 

4 
5 

3     is  i 

6 

4     is  \ 
6     is  J- 

10 

82  Practice. 

Or,  consider  the  quantity  as  so  many  of  that  denominatidit 
of  money  which  is  next  higher  than  the  price,  and  take  such 
proportional  part  of  it  as  the  price  is  of  one  of  that  denomina- 
tion, 

Example  1. 

Whatcost726^&.  at  lld.^ 

Operation.  , 
726 
11 

U)r986  d, 

20)665..  6 

^33  ..  5  ..  6  Ms. 

Example  2. 

•V^Tiat  cost  734Z6.  at  46^.? 

Operation,  Here,  I  consider  734  sls 

734  s,  =  amount  at  Is.     so  many  shillings  ;  and 


4d  is  is. 
20 


since  4d,  is  ^  of  a  shil- 
244  s.  Bd.  «=4am't  at  4d.   ling,  I  take   |  of  734s. 

which  is  244s.  Sd.    for 

^12  ..  4  ..  8  Ans,  the  answer  in  shillings, 

wliich  being  reduced  to 
pounds,  gives  ^12 ..  4 ..  8,  for  the  answer. 

JS^'ote.  Ife  dividing  734  by  3,   there  is  2  remainder ;  but  because  the  diyi' 
^ead  ia-  shillings,  it  is  2  thirds  of  a  shilling,  that  is  Sd. 

Case  2, 

When  the  quantity  is  of  one  denomination,  and  the  price 
is  of  different  denominations. 

Rule. 

Multiply  the  quantity  by  the  highest  denomination  of  the 
price,  and  add  such  proportional  part  of  it,  as  the  remainder 
of  the  price  is  of  one  of  that  denomination. 

Or,  reduce  the  price  to  the  lowest  denomination,  and  work 
fey  Case  l. 

Example  1. 
What  cost  TSeib.  at  2s,  6d,l 


Practice, 

83 

§peration» 

Here,  I  first  multiply 

736 

736,  the  quantity,  by  £s. 

2 

the  highest  denomination 

of  the  price,  &  it  makes 

$d.  is  i 

1472  =  amount  at  2s. 

1472  s.,     which    is    the 

368  =  amount  at  6d. 

amount    of    the    whole 
quantity  at   2s.     Then, 

20 

1840  =  am't  at  2s.  6d. 

because  6d.  is  J  of  2s.  I 

add  i  of  1472s.  which  is 

£9'2 ,.  0  Ms. 

368s.,  and   have  1840s. 

for  the  amount  at  2s.  6d^ 

This,  being  reduced  to  pounds,  gives  ^92,  for  the  answer. 

Example  2. 

What  cost  428  tons,  at  £3  ..  4< .. 

6^  per  ton  ? 

Operaiior 

. 

^- 

4s.  is  i 

428  amount  at  1  £^ 
S 

1284  amount  at  3 

£> 

6d.isi 

85  ..  IS  amount  at  4s. 

ic/,  is  j\ 

10 ..  14  amount  at  6^. 

17..  10  amount  at  ^d. 

^1381..    3..  10  am't  at  ^3..  4..  6§,Jns. 

*  Here,  1  first  multiply  428,  the  quantity,  by  £3,  the  highest 
^denomination  of  the  price,  and  it  makes  ^i^84,  which  is  the 
amount  of  the  whole  quantity  at  £3.  Then,  because  4s.  is 
I  of  a  pound,  I  take  J  of  ^6428,  which  is  1,85  ..  12,  for  the 
amount  at  4s.  Then,  because  66^.  is  i  of  4s.  1  take  |  of 
jL85  ..  12,  which  is  LiO..  14,  for  the  amount  at  ed.  Lastly, 
because  id.  is  j\  oi6d.  I  take  Jj  of  LIO  ..^14,  which  is  1 7s. 
lOd.  for  the  amount  at  id.  All  which,  added  together,  gives 
X1381 ..  3  ..  10,  for  the  answer. 

Case  3. 
When  both  the  price  of  one,  and  the  quantity,  are  of  diffe- 
rent denominations. 

Rule. 
Multiply  the  price  by  the  highest  denomination  of  the 
quantity,  and  take  proportional  parts  for  the  rest. 

Or,  work  by  the  Rule  of  Three,  which  will  usually  be  the 
better  way. 


u 


Fractice^ 


Example. 


iVhat  C04  17  Cwt.  Sqr.  mb.  at  L2  ,.  2..  6  per  Cwtt 
Operatiorio 
£.     s.     d. 

2gr.  i«  \ 

2..    2.,  6x5                         12+5=17 
12 

25  ..  10 ..  0  amount  of  12  Cwt. 

10 ..  12 ..  6  amount  of  5  Cwt. 

l^r.  isf 

1  ..    1 ..  3  amount  of  2  qrs. 

16(6.  is  1 
2(6.  is  \ 
1(6.  is  X 

10 ..  7^  amount  of  1  qr. 
6  .•  0§  amount  of  ]  6  Ih, 
9  amount  of  2  lb. 

4J  amount  of  1  lb. 

jess  ..  1 ..  6|  amount  of  17  Cii;f.  S^rs.  19(6. 
The  reasoA  why  I  multiply  by  1^^  and  by  5,  is,  that  13  and 
5  is  17,  and  multiplying  ^82 ..  2  ..  6  by  12  and  bj  5,  and  add- 
ing their  products,  is  the  same  as  multiplying  it  by  1 7.  In 
taking  the  parts,  when  I  say  1 6(6.  is  4,  it  is  not  4  of  Iqr.  but 
^  of  1  Cwt,  and  the  upper  line  is  to  be  divided  by  7,  and  not 
the  line  opposite  to  which  it  stands,  as  in  the  rest. 

TARE  ANDTRETT, 

Are  allowances  made  to  the  buyer  on  some  particular 
commodities. 

Tare  is  the  weight  of  the  barrel,  box,  bag,  &c. 
Trett  is  an  allowance  of  4(6.  per  104(6.  for  waste  and  dust. 
Gross  is  the  weight  of  the  goods,  together  with  that  in  which 
they  are  contained. 

JS*eai  is  the  weight  of  the  goods,  after  all  allowances  are 
deducted. 

Case  1. 
When  the  tare  is  so  much  in  the  whole  gross  weigW. 

'Rule. 
Subtract  the  tare  from  the  gross,  and  the  remainder  is  the 
neat.  Example. 

What  is  the  neat  weight  of  56  Cwt.  Iqr.  19(6,  of  tobacco^^ 
the  tare  being  15  Cwt.  ^r.  IS (6.  ? 
Cwt.  qr.    lb. 
56 ..  1  ..  1 9  gross. 
15..  2..  13  tare. 


40..3r»   6  neati  Ans* 


Tare  and  Trett 


ts 


Case  2. 

When  the  tare  is  so  much  per  barrel,  box,  &c. 
Rule. 

Multiply  the  number  of  barrels,  boxes,  &c.  by  the  tare, 
and  the  product  will  be  the  whole  tare^  which  subtract  from 
the  gross,  the  remainder  is  the  neat. 

Example. 
What  is  the  neat  weight  of  i  6  hhds.  of  tobacco,  the  gross 
being  86  Cwt.  2gr.  14^6.  and  the  tare  being  lOOlb,  per  hhd.  ? 

Tare.  Cwt  qr,  IL 
16  86  ..  ^..  14  gross. 

100  14..  I  ..    4  tare. 

4)  ^ ^ 

2S)  \e0()lbs.{57qrs.  4lb.  72  ..  1 ..  10  neat,  Arts. 

140 


200 
196 


14..  1..  4 
Cwt*  qr.  lb. 


Case  S. 
When  the  tare  is  so  much  per  Cwt. 

Rule. 

Deduct  from  the  2;ross  such  proportional  part  of  it,  as  the 
tare  is  of  a  (  wt.  and  the  remainder  will  be  the  rseat 

Or,  multiply  the  pounds  gr^JbS  by  the  tare  per  Cwt.  and 
divide  the  product  by  112,  the  quotient  will  be  the  tare, 
which  deduct  as  before. 

Example. 
In  12  butts  of  currants,  each  7  Cwt.  Iqr.  lOlb.,  tare  per 
Cwt.  iQlb.,  how  much  ;ieat  ? 

Cwt.  qr.   lb. 

7..  I..  10 

12 


I6lb.  is 


88  ..  0  ..    8  gross* 
\  2  ..  2 ..    9  tare. 


75..  I  ..^7  neat,  Ans. 
Or,  according  to  the  second  method,  as  follows : 
H 


S6  Tare  and  Trett. 

Cwt  qr.  lb. 

7..  1..  IG 

12 


88..  0..    8  §864 J&.  gross. 

4  1409/6.  tare. 


352  S8)8455  lb.  neat. 

28  84         (301  qrs. 

2824  55         4)301 

704  28  

—  —  75 ..  1 

9864  i&.  gross.  27  76. 

1 6  lb.  tare  per  Ct«^. 
. JItns.  75  Cwt.  Iqr.'Sink 

59184 

9864 

112)157824(1409  i5.  tareo 
112 

458 

448 

1024 
1008 

16 

Case  4o 

When  trett  is  allowed  with  tare. 

Rule. 

Deduct  the  tare  as  before,  and  the  remainder  is  called 
suttle  ;  which,  divided  by  ^^^6,  (which  is  i  of  1 04,)  will  give 
^he  trett,  and  that  being  subtracted  from  the  sutUe,  the  re- 
ijaainder  will  be  the  neat. 

Example. 

In  BCwt.  Sqr.  QOlb.  gross,  tare  38i6.  trett  4  li.  per  104  Ib^ 
kow  man  J  lbs.  neat  ? 


Tare  and  Treit* 


$T 


€wtqr.  lb. 
8  ..  3  ..  20 
4 

55 

28 

^2  lb.  suttle, 
37  lb.  trett. 

300 
70 

925  lb.  »eat,  w 

ICGO  IL  gross. 
38  IL  tare. 

6)962  lb.  suttle. 
78        (37  lb. 

182 
182 

trett 

fc^5.  Another  allowance,  called    CloJ]  is  sometimes  made,  afQlb.  for 

t'Xery  3  C-wt.,  which  may  be  found  by  Case  3. 

Questions  on  the  foregoing. 


What  is  practice  ? 

What  is  the  first  case  ? 

What  is  the  rule  ? 

The  second  case  ?  the  rule  ? 

The  third  case  ?  the  rule  ? 

What  i;i  tare  ? 

What  is  tiett  ? 

What  is  gross  weight  i 


What  is  neat  weight  ? 

What  is  suttle  ? 

What  is  the  first  case  ?  the  rule  ? 

The  second  ?  the  rule  ?    The  third? 

the  rule  ? 
The  fourth  ?  the  rule  ? 
Why  do  you  divide  by  26  ? 


INTEREST, 

Ts  an  allowance  made  fer  the  use  of  money. 

The  Frincipal  is  the  sum  at  mterest,  or  the  sum  for  the 
use  of  which  the  allowance  is  made. 

The  Rate  per  cent  is  the  interest  of  j8iOO,  or  glOO,  or  the 
allowance  made  for  the  use  of  it,  for  one  year. 

The  Amount  is  the  sum  of  the  piincipal  and  interest. 

SIMPLE  INTEREST, 

Is  that  v/hlch  arises  from  the  principal  only. 

Case  1. 
To  find  the  interest,  when  the  principal,  time,  and  rate  per 
cent,  are  given. 

Rule  1. 
Say  :  As  100  is  to  the  rate  per  cent, 

So  is  the  principal  to  the  interest  for  one  year.  Tlien, 


S8  Simple  Interest. 

Multiply  the  principal  bj  the  rate  per  cent,  and  divide  the 
product  bj  100.  The  quotient  will  be  the  interest  for  one 
year. 

JK'otel.  To  divide  by  100,  point  off  the  two  right  hand  figures,  and  the 
other  figfures  will  be  the  quotient. 

JVote  2    To  multiple  by  a  mixed  number,  as  6  J,  or  5§,  multiply    by    the 
Tfhole  number,  and  take  proportional  parts  for  the  fraction. 
Example  1. 
What  is  the  interest  of  £S7 ..  14  ..  5  for  one  year,  at  6  per 
eent  ? 

Here,  I  multiply  the  principal,  LS7 ..  14  ..  5, 
by  the  rate  per  cent,  6,   and  the  product  is 
JL526  ..  6  ..  6.     The  pounds  being  divided  by 
icO,  give  L5  interest,  and  L-6  remainder. 
I  set  down  L .  in  the  answer,   and  multiply 
the  26  by  9.0,  to  bring  it  into  shillings,  adding 
in  the  6  shillings,  and  it  makes  526  shillings. 
This,  again^  being  divided  by  ?  00,  gives  5s. 
interest,  and  26s.  remainder.     I  set  down  5s, 
in  the  answer,  and  multiply  the  26  by  12,  to 
bring  it  into  pence,  adding  in  the  6  pence,  and 
it  makes  318  pence.     This  again  being  divi- 
ded by  100,  gives  3  J.  interest,  and  \Sd,  re- 
mainder.    1  set  down  Sd,  in  the  answer,  and 
5     multiply  the  18  by  4,  to  bring  it  ml  o  far- 
things,  and  it' makes  72  farthings,  which 
being  less  then  100,  cannot  be  divided  by  it,  and  makes  only 
y\2_  of  a  farthing,  and  is  of  no  importance  in  this  place.     So 
that  the  answer  is  L5  .^5  ..  3. 

Becond  method.  Retluce  the  principal  to  its  lowest  deno- 
mination ;  then  multiply  by  the  rate  per  cent,  and  divide  by 
100  ;  and  then  reduce  the  answer  to  a  higher  denomination. 
Take  the  same  example  : 
L.     s.     d. 
ST..  14..  5 
20 

1754, 
12 

21053  principal  in  pence 
6  rate  per  cent* 

12  63- 18. 


87..  14 

d. 

..  5 

6 

5'£6..    6 
20 

..6 

5-26 

12 

3*18 

4 

•72 
Jins.  L5 .. 

5.. 

Simple  Inierest  S^ 

The  interest,  therefore,  is  12636^.,  which  being  reduced  to 
pounds,  is  L5  ..  5  ..  3,  as  before. 

Example  2, 

What  is  the  interest  of  S37-45  for  one  year,  at  5|  per 
cent  ? 

837*45  This   product  being  divided    by 

5  J  100,   or  the  decimal  point  being  re- 

moved  two  figures  to  the  left,  is  £•- 

187-25  product  by  5.     597^,.  or  2  dollars,  5  cents,  9  mills, 

18«72§  product  by  J.  and  7§  tenths  of  a  mill,  which  is  the 

answer, 

205-97^  product  by  5^. 

EXAMPLE  3. 

What  is  the  interest  ot  X£0 ..  16  ..  6  for  one  year,  at  6| 
per  cent  ? 

L.     s.    d.' 
20..  16..6 
6| 


124..  19..  0  product  by  6. 
10..    8..  3  product  by  §. 
5  ..    4  ..  1 J  product  by  i* 

1-40  ..  11 ..  4j  product  by  6^. 
20 

—  Ms.  L\  ..  8..  li. 
8-11 

12  Having  found   the  interest   for  one 

—  year,  find  it  for  any  other  time,  bymul- 
1*36  tiplication,  the  rule  ofthree,  orpractice, 

4     as  the  case  may  require. 

1-46 

Rule  2. 

i"  To  find  the  interest  for  any  time  at  6  per  cent,  multiply 

the  principal  by  half  the  time  in  months,   and  divide  by  1 00, 

Example  1. 

What  is  the  interest  of  §9 1 6'72  for  one  year  and  4  months^ 

at  6  per  cent  ? 

Operation, 
g9 16*72  prmcipaL 

8  half  the  number  of  months. 

So  the  answer  is  73  dt)llars,  SS 

7Z'^376    cents,  7  mills,  and  j\  of  a  inilL 

H2 


90  Simple  Interest. 

Example  2.  ^ 

What  is  the  interest  of  1,342 ..  1 6  .•  6  for  one  year  and 
months,  at  6  per  cent  ? 

£.     s.     d. 
S42..  16..  6 

7|  half  the  number  of  months. 


2399  ..  15  ..  6  product  bj  7. 
171  ..    8..  3  product  by  j. 

25-7 1  ..    3  ..  9  product  by  74. 
20 

14-23 
12 

—  Ans.  ^25..  14..  2 J. 

2-85 
4 

S-4(> 

Rule  3. 
Multiply  the  principal   by  the  number  of  days,  and  that 
product  by  the  rate  per  cent,  and  divide  by  36500  y  the  quo- 
tient will  be  the  interest. 

Example. 
What  is  the  interest  of  752  dolls,  for  101  days,  at  7  per 
cent  ? 

Ojjeration.     752x101=75952,   and   75952x7=531664  j 
and  53  .664-r-S6500=S  14-366+,  which  is  the  interest. 

JSTote  1.  The  reason  for  dividing  by  36500,  is,  that  there  are  365  days  in  a 
year,  and  dividing  by  36500  is  the  same  as  dividing  by  365  and  by  100. 

J\ote  2.  Instead  of  dividing  by  36500,  you  may  multiply  by  the  decimal 
'0000274,  T/hieh  is  f  nearly  J  the  quotient  of  1  divided  by  36500  ;  and  the 
result  will  be  the  same.  Or,  in  Fe^leral  moneys  multiply  by  274,  and  point 
off  the  seven  right  hand  figures  for  decimal  parts  of  a  dollar.  Take  the  same 
question  : 

752x101=75952,  and  75952x7=531664,  and  53 1664 x 
•0000274=1  i-5675936,  or  S 14-5675936,  which  is  the  interest- 

Case  2. 
The  method  pursued  by  the  banks. 
As  the  interest  of  1 00  dollars  at  6  per  cent  is  just  1  dol- 
lar, or  100  cents,  for  two  months,  the  interest  of  any  number 
of  dollars  for  that  time,  is  the  same  number  of  cents.    Hence 
the  following 


Simple  Interest  9 1 

Rule, 
t  down  the  principal  in  dollars  and  decimal  parts  of  a 
dollar,  and  remove  the  decimal  point  two  figures  to  the  left, 
and  you  have  the  interest  of  that  sum  for  two  months  at  6  per 
cent.  For  any  greater  or  less  time,  work  by  multiplicatioa^ 
or  practice. 

IR  JSTote  1    If  you  wish  to  find  the  interest  at  7  per  cent,  add  to  the  interest  at 
5  per  cent,  I  sixth  of  itself 

JVote  2   The  banks  reckon  30  days  a  month,  and  360  days  a  year  ;  so  that 
by  this  method  tliey  gain  tlie  ii^terest  of  5  days  in  a  year. 

Example   i.. 
What  is  the  interest  of  Dr856'43  for  3  months,  at  7  per 
eent? 

Operation. 
6)78'5643  interest  for  2  months  at  6  per  cent* 
13-09405  i  to  be  added. 


2)9  1*65835  interest  for  2  months  at  7  per  cent. 
45*82917  interest  for  l  month  at  7  per  cent. 

8137*48752  interest  for  3  months  at  7  per  cent,  t^ns. 

Example  2. 
What  does  a  bank  gain  in  a  year,  by  pursuing  the  above 
method,  on  500000  dollars,  at  7  per  centB 
Operation. 

6)5000-00  int.  for  60  days  at  6  p.  c> 
833-33J  I  to  be  added. 


5  days  is  J3  of  60  days...t2)5833-33j  int. for  60  days  at  7  p.  c» 


•11-f-  Ans. 

Case  3. 
To  find  the  sum  due  on  an  obligation,  when  there  are  se- 
veral payments* 

Rule  1. 

1.  Find  the  amount  of  the  principal  for  the  whole  time.^ 

2.  Find  the  amount  of  each  payment,  computing  the  inte- 
rest on  it  from  the  time  it  was  made  to  the  time  of  settle- 
ment. 

3.  Subtract  the  amount  of  all  the  payments  from  the 
amount  of  the  whole  sum,  and  the  remainder  will  be  the  sum 
due. 


9x!  Simple  Interest*' 

Example. 

A  gave  a  note  to  B,  dated  Jan.  1,  1780,  for  8^000,  payable 
on  demand,  with  interest  at  6  per  cent,  on  which  are  endorsed 
the  following  payments  : 

i.  April  1,  1780,         S2^ 


2.  August  1,  1780,  4, 


What  is  due   on  this  note, 
June  1,  1784? 


3.  December  1,  1780,     6. 

4.  February  1,  1731,     60. 

5.  July  1,  1781,  40. 

Operation. 

1.  The  whole  time  from  Jan.  1,  1780,  to  June  1,  1784,  is 
4  years  and  5  month-.  The  interest  of  S  000  for  that  time, 
is  g26o  ;  consequently  the  amount  is  %\2C5, 

2.  The  first  payment  is  8:^4.  It  was  made  April  1,  ITSO, 
From  which  to  the  time  of  settlement,  is  4  years  and  2  months. 
The  interest  of  824  for  that  time,  is  g6,  which  added  to  24, 
the  amount  of  the  first  payment  is  gSj. 

3.  The  second  payment  is  84,  made  Aug,  1,  1780.  Its 
time  is  3  years  and  lo  months  5  and  its  interest  92  cents, 
which  makes  its  amount  84-92. 

4.  The  third  payment  is  86*  made  Dec.  1,  1780.  Its  time 
is  3  years  aiad  6  months,  and  its  interest  Sl*26,  which  makes 
its  amount  87-26. 

6.  The  fourth  payment  is  860,  made  Feb.  1,  1781.  Its 
time  is  3  year^j  and  4  months,  and  its  interest  81^/  which 
makes  its  amount  872. 

6.  The-fifth  payment  is  840,  made  July  1, 1781.  Its  time 
is  Z  years  and  II  months,  and  its  interest  87,  which  makes 
its  amount  8 '7. 

7.  The  whole  amount  of  the  payments,  then,  is  8 161-18; 
which  being  subtracted  from  the  amount  of  the  whole  sum, 
gl265,  gives  the  sum  due  Si  103-82,  which  is  the  answer. 

Rule  2. 

The  method  established  by  the  courts  of  law  in  Massachu- 
Beit:-,  is  the  following  : 

Cast  the  interest  on  the  whole  sum  up  to  the  time  of  the 
first  payment;  and  if  the  payment  exceeds  the  interest  then 
due,  deduct  that  excess  from  (he  principal,  and  con-ider  the 
remainder  as  thfe  new  principal,  and  cast  the  interest  on  that 
up  to  the  time  of  anotlier  payment,  and  so  on.  But  if  the 
first  payment  does  not  exceed  the  interest  due  when  it  was 
made,  cast  the  interest  on  the  whole  sum  from  the  date  of  the 
W)ligatiQn  up  to  the  second  payment,  and  see  if  the  first  and 


I  Simple  InterrsU  9S 

second  payments,  taken  together,  exceed  the  interest  due  at 
the  time  of  the  second  payment.  If  they  do,  deduct  their 
excess  from  the  principal,  as  before ;  but  if  they  do  not,  cast 
the  interest  again  upon  the  whole  sum  up  to  the  time  of  the 
third  payment,  or  the  fourth,  or  till  such  time  as  the  payments 
taken  together  do  exceed  the  interest  then  due,  and  then  de- 

^   duct  as  before. 

I      Take  the  preceding  example. 

f  Operation. 

1.  From  Jan.  1, 1780,  the  date  of  the  note,  to  April  1, 1 780, 
the  time  of  the  first  payment,  is  3  months.  The  interest  of 
glOOO  for  ^^  months  at  6  per  cent,  is  Dl5.  The  payment 
made  at  that  time,  1)24,  exceeds  the  interest  then  due  by  D9, 

J    which  is  therefore  to  be  subtracted  from  the  principal,  which 
being  done,  leaves  D991,  to  form  the  new  principal. 

2.  From  April  1,  1780,  the  time  of  the  first  payment,  to 
Aug.  1,  1780,  the  time  of  the  second,  is  4  months.  The  in- 
terest of  D991  for  4  months  is  D19-82.  The  payment  then 
made  is  D4,  which  is  not  so  much  as  the  interest  then  due. 
So  there  is  nothing  to  be  deducted  from  the  principal  at  that 
time. 

3.  The  third  payment  is  made  Dec.  1,  1780,  But  as  the 
second  payment  did  not  exceed  the  interest  due  at  the  time 
it  was  made,  I  go  back  and  compute  from  April  1,  (780,  the 

.   time  of  the  first  payment,  to  Dec.  I,  1780,  which  is  8  months. 

The  interest  of  D99]    for  8  months  is  D39-64.     The  third 

payment  is  D6  ;  and  this  added  io  Dl,   the  second  payment, 

\  is  DlO,  which  is  less  than  the  interest  due  at  the  time  of  the 

;   third  payment.     So,  there  is  no  deduction  to  be  made  at  that 

time. 

4.  The  fourth  payment  is  made  Feb.  1,  1781.  But  as  the 
second  and  third  payments  did  not  exceed  the  interest  due 
at  the  time  when  they  were  made,  1  must  still  go  back,  and 
compute  from  the  time  of  the  first  payment,  April  1,   1780. 

;  From  that  time  to  Feb.  1,  178i,  is  10  months.     The  interest 

|ofD991   for    10  months  is  D49*55.     The  fourth  payment  is 

V  D60,  which,  with  the  two  preceding,  D4    and  D6,   is  D70. 

'  This  exceeds  the  interest  now  due,  by  D20-45  ;  which  being 

deducted  from  D991,  the  principal,  leaves  D970'55,  for  the 

w  principal. 

5.  The  fifth  payment  is  made  July  1,  1781,  which  is  5 
months  from  the  time  of  the  fourth.  The  interest  of  D970'55 
for  5  months,  is  D24*£6 ;  and  the  payment  is  D40,  which  ex- 


94  Simple  Liter  est 

eeeds  the  interest  then  due  by  Di5-r4,  and  that  being  de- 
diu  ted  from  1)970^55,  the  principal,  leaves  Do  .'Si,  i&r  the 
new  principal. 

6.  Froir  the  time  of  the  fifdi  payment  to  June  1,  1784,  the 
tiir.8  yif  settlemeat,  is  ':  ye  rs  and  1.  '-  months.  The  interest 
of  \)  ■  -b-  i  for  2  year:3  .md  1  months,  is  D; 67*09  ;  which 
being  added  to  the  priacipid,  makes  the  sum  due  D)i2l*90, 
Vv  h  1  c  h  is;  t lip  a  n  ^ w  e r. 

JYote.  The  medu/^l  "  ~  '  rul  ■  is  ntore  fir  orableto  the  (L  bfor,  as  ap- 
peft  s  b}  t  e  foreji-oi(, .  he  .';  t^    enct'  in  ihe  Hip.oiint  being  §18  08. 

But  'he  mrthdd  or  r:  i  ;^  eq-iitnble.     B(  cause   it   u&y  so 

ha'M^en,  h)  xlnt  iii':"*  h^  ,    ..liKt    i'l  a  course  of  yetirs,  ihe  obli'- 

gat'on  niMY  be  tancc..  hj'idcr  (A'  .h  ^  note  b..-';ught  in  debt  ta 

tii<   ;>;iver,  by  tiie  pa}  :r^'    t  ,  st  oni) ,    '.vr)i  -ut  any  part  of  the  priii- 

ti:-:.!  being  paid  ;  :-3  apiior.  1  ov\ing  qu«^stioii :  Suppose  a  liote  was 

give.!!  Jan    ■     5  800,  fm^  ':ii'.  "t  at  6  per  ce.  t,  and  thil  f^5  a  year  is 

pr.    !  on    .he  Rrot  of  J  year,  till  Jan.  1,  1850;  how  does  ths 

ace   ii   I  o.  n  jitand  a  st  ruJe  ' 

ih<^  pri;;cipa'  is  ^3^**^'  '  ''  '•-''■''  -f^nst  on  it  N-,6  a  year,  for  50  years,  is 
D30v> ;  <o  that  ;he  a'.uount  •-'iTn  j:  -  cipai  is  DiOO".  1  h.  r.^  at  49  pa)  .nents 
of  6  doliurs  e  n-h.  lee  inter  st  ci  ilc  first  ffr  49  \  ears,  is  D5  7-64,  and  its 
aPi'MiriV  is  ]>J.,^  G4.  i  h.^  Juterest  of  the  teco'  d  for"48  \ears  is  D17-28,  and 
its  auioiint  Is  I)'23  t28  and  so  on  F.  ora  compitUni^  which,  it  appe.irs,  that 
tlie  ;»nciOu'.t  rif  di  several  pa  tnents.  viJh  the  inte'  est  on  them,  is  't>7.'35.  So 
that  at  the  time  of  s:tt!einent,  <he  hold' r  of  the  note  is  indebted  to  the  giver 
of  it  I):>35.  \Vh  rea  ,  in  equit;' ,  n<  thing  has  been  paid  but  the  interesCas  it 
pr  pe'  i-    bo.  a  ,,:-  ^^  :•■,  9i;o  )io;>e  of  th-   principal  l;as  ever  been  paid, 

'i  ';f  .  CO;.'!  r  'le  ;s  |!erh;r  s,  as  equitable  as  Chn  cor.veiieiitly  be  made,  but 
is  not  e:\'-tC(ly  so  Because  the  interest  '>iight  to  b<:^  paid  at  the  nd  of  every 
year,  i^  it  is  paid  sooner,  there  is  an  advantage  to  the  creditor  ;  if  later,  to 
the  debtor. 

•   Case  4. 

To  find  the  rate  per  cent,  when  the  amount,  time,  and 
principal,  are  given. 

Rule. 

As  ihe  principal  is  to  the  interest  for  the  whole  time,  so  h 
LlOO  to  its  interest  for  the  same  time.  Having  found  the 
interest  of  jL  00  for  any  given  time,  the  interest  ef  it  for  ons 
year  may  be  toun  I  by  division,  multiplication,  or  practice,  as 
circumstances  recpire. 

'  Example. 

At  what  rate  per  cent  will  X500  amount  to  L725  in  9 
years  ? 

Here,  I  wish  first  to  find  the  interest;  and  knowing  the 
principal  and  the  amount,  I  subtract  the  principal,  1*500, 
from  the  amount,  X725,  and  the  remainder,  L225,  is  the  in- 
terest.    1  then  say  : 

As  L500  is  to  L2'^i5,  so  is  L.OO  to  its  interest  for  9  years. 
This  proportion  being  worked  out,  gives  X45  as  the  interest 


Simj)le  Interest  95 

«f  jLIOO  for  9  years  ;  and  this  being  divided  by  9,  gives  L5 
as  the  interest  of  L 100  for  one  year,  that  is,  5  per  cent,  which 
is  tiie  answer. 

Casf.  5. 
To  find  the  time,   when  the  principal,  amount,  and  rate 
per  cent,  are  given. 

Rule. 
Divide  the  whole  interest  by  that  of  the  principal  for  one 
year,  and  the  quotient  will  be  the  time. 
Example. 
In  what  time  will  7>500  amount  to  L79.5^  at  5  per  cent  ? 
Here,  I  first  find  the  interest  of  jL500  for  one  year  at  5 
per  cent,  which  is  L25  ;  and  then  I  find,  by  subtracting,  as 
before,  what  is  the  whole  iHt^rest,  which  is  L225  ;  and  then 
I  divide  L225  by  25,  and  it  gives  9  years  as  the  answer. 
Case  6  — Discount. 
To  find  the  principal,  when  the  amount,  time,  and  rate  per 
«ent,  are  given. 

This  case  is  the  same  as  the  rule  of  Discount. 
Discount  is  an  abatement  made  for  the  payment  of  money 
before  it  becomes  due,  by  accepting  as  much  as,  being  put 
©ut  at  interest,  would  amount  to  tlie  whole  debt  at  the  time 
it  becomes  due. 

The  present  worth  is  the  amount  so  accepted ;  and  is  the 
Same  as  the  principal  found  by  this  case. 

Rule. 
As  the  amount  of  LlOO,  at  the  rate  and  time  given,  is  to 
XlOO,  so  is  the  amount  or  whole  debt  to  the  principal  or 
|)resent  worth. 

,       To  find  the  discount,  subtract  the  present  worth  from  the 
whole  debt,  and  the  remainder  will  be  the  discount. 

Proof. 
Find  the  amount  of  the  present  worth  at  interest  for  tlie 
same  rate  and  time,   according  to  Case  1,  and  it  will  equal 
the  whole  debt,  if  the  work  is  \:ight. 

Example.     ; 
What  is  the  discount  of  L400  for  6  months,  at  6  per  cent  ? 

(hp^eration. 
Here,  I  first  find  what  LlOo  will  amount  to,^t  interest  for 
6  months  at  6  per  cent,  which  is  L  03.  1  then  say,  as  LIOS 
is  to  LlOO,  so  is  L^OO  to  the  present  worth  ;  which  propor- 
tion being  worked  out,  gives  jLS88  ..  6 ..  I IJ,  for  the  present 
worth  ;  which  being  subtracted  from  X40<%  the  whole  debt, 
gives  1/ 1 1 ..  13 ..  Oi,  as  the  discount,  which  is  the  answer. 


\ 


S6  Compound  Interest. 

Insurance,  Commission,  and  Brokage, 
Are  allowances  made  to  insurers  and  factors,  or  other 
agentSj  at  a  stipulated  rate  per  cent ;  and  the  amount  of  such 
allowance  is  the  same  as  the  simple  interest  for  one  jear  at 
the  same  rate  per  cent,  and  is  found  in  the  same  manner. 


COMPOUND  INTEREST, 

Is  that  which  arises  from  a  principal  increased  from  time  to 
time  by  the  addition  of  the  interest  to  the  principal,  as  often 
as  the  interest  becomes  due. 

Rule. 

Find  the  first  year's  amount  by  Simple  Interest,  which  will 
be  the  principal  for  the  second  year  ;  and  the  amount  for  the 
second  year  will  be  the  principal  for  the  third  year,  and  so 
on.  From  the  last  amount,  subtract  the  first  principal,  and 
the  remainder  will  be  the  compound  interest. 
Example. 

What  is  the  compound  interest  of  jL450.  for  3  years,  at  5 
per  cent  ? 

Here,  I  first  find  the  interest  of  La^50  for  one  year  at  5  per 
cent,  which  is  i22 ..  i  0.     I  then  add  this  to  the  principal,  and 
the  amount  for  one  year  is  L-^7'' ..  10,  which  is  the  principal 
for  the  second  year.     1  then  find  the  interest  of  L47'^->  10 
for  one  year,  which  is  I>23  ..  12  ..  6  ;  and  this  being  added  to  , 
its  principal,  the  amount  for  the  second  year,  is    496  ..  2  ..  6, 5 
which  is  the  principal  for  the  third  year.     I  then  find  the  in-  ' 
terest of  M96 ..  2 ..  6  for  one  year,  v/hich  is  X2 i ..  16..  IJ  ; 
and  this  being  added  to  its  principal,  the  amount  for  the 
third  year  is  jL:'20  ..  18  ..  7|.     Then  subtracting  the  or'ginal 
principal,  M50,  from  this  last  amount,  1  have  LjQ„  18 ..  7^,  ^ 
as  the  compound  interest,  and  answer. 

JVote.  In  this  example,   the^ interest  is  supposed  to  be  pa\  able  annually. 
If  it  is  pa^able  more  or  less  frequently,  the  iiteiesi  must  be  calculated  up  to  : 
the  time  when  it  is  due,  and  then  added  to  its  principal,  to  foi^ffi  a  new  prin«  \ 
cipal. 

Questions  on  the  foregoing. 

What  is  the  second  method  of  find* 
ing  thfc  interest  r 

Having  fend  the  interest  for  one 
yeai ,  how  do  you  find  it  for  more 
than  a  j  ear  ? 


What  is  interest  ? 

What  is  the  principal  ? 

"What  is  the  rate  per  cent  ? 

What  is  the  amount  ? 

Wliat  is  simple  interest  ? 

What  is  th^  fifst  case  ? 

What  is  the  first  rule  ? 

How  do  you  divide  by  100  > 

How  do  >  ou  mvdtiply  by  ft  mixed 


How  for  less  than  a  year  ? 

What  is  the  second  rule  f 

What  is  the  third  rule  ?  f 

Why  »io  you  divide  by  36500  ? 

Wiuit  ia  ^«  f¥«oAa  ci^e  f  the  iruW  t 


Exercises, 


^7 


What  is  the  thu*d  case?  the  first 
rule  ?  the  second  ? 

Which  is  the  more  favorable  to  the 
debtor  ?  Which  is  the  more  equi- 
table f 

What  is  the  fourth  case  ?  the  rule  ? 

The  fifth  case  ?  the  rule  ? 

The  sixth  case  ?  W  hat  is  discount  ? 
What  is  the  present  worth  ? 

How  do  you  find  the  present  worth  ? 
How  the  discount  ? 


How  do  you  prove  the  operation  ? 
What  are   insurance,     commission, 

and  brokage  ? 
How  are  they  found  ? 
What  is  compound  interest  ? 
What  is  the  inile  ? 
When  you  have  found  the  amount 

for  any  given  time,   how  do  you 

find  the  Qompound  interest  ? 
What  if  the  interest  is  not  payable 

annually  ? 


Exercise  .^0. 


Tell  the  interest,  at  6  per  cent,  jins. 
Of 81*^0      for    12 months,  8900 

8-00 
21  00 
IfrOO 

3-00 
22-50 

7-50 
13-50 

6-00 
10-00 


Tell  the  interest,  at  6  per  cent. 
Of  8250  for    6  months, 


200 

8 

700 

e 

800 

4 

600 

1 

750 

6 

100 

15 

300 

9 

400 

3 

600 

4 

125 

225 

275 

300^ 

350 

400 

475 

425 

650 


12 
6 
6 
4 
6 
8 
6 

12 
4 


7-50 

7-50 

6-75 

8-25 

600 

10-50 

16-00 

14-25 

25-50 

1100 


Exercise  51. 


Tell  the  interest,  at  5  per  cent,  Ans, 
Of  gl20  for  12  months,  D600 
2-50 
6-25 
4-25 

11-50 
7-25 

20-00 
6-25 

16-50 
7-75 


Tell  the  interest,  a(  5  per  cent.    Am. 
Oft)7C0  for    6  months,     Dl7-oO 


2(i0 

3 

250 

6 

340 

3 

4€0 

6 

580 

3 

600 

8 

500 

3 

440 

9 

620 

3 

140 
260 
380 
900 
860 
180 
240 
560 
640 


3-50 

9-75 

4-75 

1500 

21-50 

4-50 

7-00 

7-00 

24-00 


EQUATION, 

Is  the  reduction  of  several  stated  times  at  which  money  is 
payable,  to  one  time,  which  shall  be  equal  in  value. 

Rule. 
Multiply  each  payment  by  its  time,  and  divide  the  sum  of 
all  the  products  by  the  whole  debt,  the  quotient  will  be  the 
mean  or  equated  time. 

Proof. 
The  interest  of  the  sum  payable  at  the  equated  time,  will 
be  equal  to  the  intexest  of  the  several  payments  at  their  re^ 
spective  times. 

I 


^8  Barter.-^Loss  and  Gain, 

Example. 
A  owes  B  LI 00,  of  which  i^50  is  payable  in  two  months, 
and  L50  at  four  months  ;  what  is  the  equated  time  ? 

O^eratwn. 
50x2=100  Here,I  multiply  each  payment 

50x4=200  by  the  number  of  months  be- 

fore    which    it  becomes  due, 

100)300(3  months,  Jlvs.      and  add  their  products,  which 

makes  300.  T  then  divide 
this  sum  by  the  whole  debt,  and  the  quotient  is  3  months,  for 
the  equated  time. 

JVote  This  is  the  common  method,  hut  it  is  not  exactly  equitahle,  hecause 
the  interest  is  allowed  instead  of  the  discount  on  the  payment  which  is  made 
before  it  would  faU  due. 


BARTER, 
Is  exchanging  one  commodity  for  another,  by  duly  pro- 
portioning their  quantities  and  values. 

Rule. 
Work  by  multiplication,  the  rule  of  three,  or  practice,  as 
occasion  requires. 

Example. 
How  much  sugar  at  ^d,  per  Ih,  must  be  bartered  for  6|  Cwt 
of  tobacco  at  lAd,  per  Ih.  ? 

Operation. 
Here,  1  first  find  the  amount  of  the  tobacco  at  l  Ad.,  which 
is  I0l92ei. ;  and  then  find  how  much  sugar  at  9d,  that  sum 
w^ill  buy,  and  it  is  10  CwL   12.1b,  and  |^  of  a  ^6.  which  is  the 
answ^er. 


LOSS  AND  GAIN, 
Is  a  method  of  computing  the  profit  or  loss  on  the  purchase 
and  sale  of  goods. 

Rule. 
"Work  by  the  rule  of  three,  or  practice,  as  occasion  requires. 

Example. 
Bought  9  Cwt  of  cheese  at  i^2 ..  16  per  Cwt.,  and  retailed 
it  at  7d,  per  lb, ;  what  is  gained  or  lost  in  the  whole  ? 
Operation. 
I  first  find  how  much  was  paid  for  the  cheese,  which  was 
Z^5 ..  4 ;    and  then  how  much  was  received,  which  was 
jL29  ..  8 ;  and  the  gain  is  i>4  ..  4. 

J^'ote.  If  the  gain  or  loss  per  cent  is  required,  it  is  found  by  the  rule  of 
nree,  as  follows  :  Make  the  sum  of  money  employed,  tlie  first  term;   the 
gain  or  loss,  the  second  ;  aad  100,  the  third.     Thus,  in  the  preceding  ex- 
ample, the  sum  employed   was  251.  is  ,  and  the  gain  was  Al.  4^.,  which  K 
r;#UT)d  to  be  W.  ids..  Ad.^^T  lOOl.  or  IQ  and  2  thards  per  eeaC* 


Fellowshijj.  99 

FELLOWSHIP, 

Is  the  rule  for  adjusting  the  several  sliares  of  gain  or  loss 
in  any  joint  business. 

Case  1. 
When  the  stocks  of  the  several  partners  continue  for  the 
gam«  time. 

Rule. 
As  the  whole  sum  or  stock  is  to  the  whole  gain  or  loss,  so 
is  each  partner's  share  of  stock  to  his  share  of  the  gain  or  loss. 

Proof. 
The  sum  of  the  several  shares  must  be  equal  to  the  whole 
gain  or  loss. 

Example. 
A  and  B  bought  a  parcel  of  goods,  for  which  A  paid  LS, 
and  B  LT,     The  goods  being  sold,  there  was  again  of  25s. 
What  is  the  share  of  each  ? 

Operaiion. 
I  first  add  the  stocks,  and  find  the  sum  to  be  LlO;  and 
tlien  say :  As  LlO  is  to  25s.,  so  is  Ls  to  A's  share,  and  L7  to 
B's  share  ;  which  proportions  being  worked  out,  give  A  7s> 
M,,  and  B  17s.  6d. 

Case  2. 
When  the  stocks  continue  unequal  terms  of  time. 

Rule. 
Multiply  each  man's  stock  by  its  time  ;  then,  as  the  sum  of 
the  products  is  to  the  whole  gain  or  loss,  so  is  each  particular 
product  to  its  share  of  the  gain  or  loss. 
Example. 
A  and  B  traded  as  follows  :  A  put  in  L50  for  6  months^ 
and  B  L75  for  3  months,  and  they  gained.  1,25..    What  is  the 
hare  of  each  ? 

Operation,. 
Here,  I  first  multiply  A^s  stock,  L50,  by  its  time,  Gmonths, 
and  A's  product  is  300.  Then  I  multiply  B's  stock,  L75, 
by  its  time,  3  months,  and  B's  product  is  225.  The  sum  of 
these  products  is  525.  I  then  say,  as  525  is  to  L'io,  so  is 
300  to  A's  share,  and  225  to  B's  share ;  which  proportions 
being  worked  out,  give  A  L14 ..  5  ..  8^,  and  about  half  a  fiir- 
thiug  ;  and  B,  LiO  ..  14..  3^",  and  about  half  a  farthing  ; 
which  is  the  answer. 


EXCHANGE, 

Is  the  rule  by  which  the  money  of  one  state  or  country  ii> 
brought  into  that  of  ano<her^ 


i^  Exchange. 

Par  is  equality  of  value.  But  the  course  of  exchange  is 
frequently  above  or  below  par. 

Case  1. 

To  reduce  the  pounds,  shillings,  pence  and  farthings  of  the 
various  currencies,  to  Federal  money. 

PtULE. 

1 .  Reduce  the  given  sum  to  pounds  &  decimals  of  a  pound, 

2.  Multiply  the  given  sum  by  the  number  of  pence  in  a 
pound,  (240,)  and  divide  the  product  by  the  number  of  pence 
a  dollar  is  worth  in  the  given  currency.  The  quotient  will 
be  dollars  and  decimals  of  a  dollar. 

Example. 

In  X250  ..  \5  ..  6  New -York  currency,  how  many  dollars  ? 

Operation'  X^aO-rro  x 240 =601 86-000  ;  and  60186--r-96, 
(because  a  dollar  is  96d.  N.  York,)  is  D626-9375,  which  is 
the  answer. 

»A^ote  1.  To  reduce  dollars  and  decimals  of  a  dollar,  to  pounds  and  deciraak 
of  a  pound»  reverse  the  operation  ;  that  is,  multiply  by  the  number  of  pence 
a  dollar  is  worth,  and  divide  by  240. 

J^^ute%  '1*6  shorten  the  operation,  divide  the  number  of  pence  a  dollar  ia 
worth,  aud  the  number  of  pence  iu  a  pound,  by  any  number  that  will  divide 
both  without  a  remainder,  and  make  use  of  the  quotients  so  found  for  amui- 
lipiier  and  divisor. 

According  to  the  above  rule,  the  following  table  is  constructed  ; 

To  reduce  dollars  to  Z.  cur- 
rency. 

rmdtiply  and  dl- 

by  tide  by 

N'ew-York,  ke.              5                    2'     New-York,  U^,              2  '        5 

.Xew-Eni^'and,  Sec.      10                     3      New-Eng'and,  &c.        3  10 

Pennsylvania,  kc.        ,8                     3      Pennsylvania,  Jkc.          3  8 

S.  Carolina,  kc.           30                    7      S.  Carolina,  kc.             7  30 

British  America,           4                    I      British  America,           1  4 

Sterling,                        40                    9      Sterling-,                          9  40 

Irish,                            IGO                  39   '  Irish,                              39  160 

To  reduce  Sterling  to  BoUars  by  a  shorter  method. 
Rule. 
Divide  the  pounds  and  decimals  of  a  pound  by  3,  and  that 
quotient  by  3  ;  add  the  two  quotients  together,  and  remove 
4\e  decimal  point  one  figure  to  the  right 
Example. 
In  I.' 1000  sterling,,  how  many  dollars  ? 

Operation^  This  is  only  an  abrido;nient 

3)lo00  of  the  other  method  ;  tor,  J 

is  |,  and  J  of  J   is  i,  and  J 

3)333-3333+  and  .V   is  J  ;    aad  removing 

1 1  M  lil  -{-  the  decimal  point  one  figure 

to   the  right,  is  the  same  as 

D4444'444+  dm^  multiplying  by  1 0 ;  and  mill- 


To  reduce  L.  cun-ency   to 
dollars. 


tiplying  by  10  and  bj  4,  is  the  same  as  multiplying  by  40  ; 
therefore/ the  two  parts  of  the  operation  are  equivalent  to 
taking  V»  cr  multiplying  by  40  and  dividing  by  9,  according 
to  the  above  table  ;  which  is-  equivalent  to  multiplying  by 
^40,  the  pence  in  a  pound,  and  dividing  by  54,  the  pence  a 
dollar  is  worth  in  sterling. 

Case  2. 
To  reduce  the  Currency  of  one  State  to  that  of  another. 

Rule  1. 
As  the  number  of  pence  in  a  dollar  in  one  state,  is  to  the 
number  of  pence  in  a  dollar  in  the  other;  so  is  tlie  numbei 
of  pounds,  &c.  in  the  ozle,  to  the  number  of  pounds^,  &c.Jii 
the  other.     Therefore, 

Multiply  the  given  sum  by  the  number  of  pence  in  a  dol- 
lar in  the  currency  requireil,  and  divide  by  the  number  ot 
pence  in  a  dollar  in  the  given  <:urrency. 

jKlte.  The  operation  may  be  shortered^  by  first  dividing  the  muUiplier  and 
diviicr  by  a  commou  divisor,  as  before  stated. 

EaAMPI.E. 

In  £l00  N.  York  currency,  how  much  S.  Carolina? 

Operation.  96  :  56  :  :  100  :  ^ns.  Therefore^ 
100x5D«=5600,  and  56C0-T-96==i>58-33SJ,  which  is  the  an- 
swcr. 

Or,  96  and  56  may  first  be  divided  by  the  common  divisor 
8,  and  th  ir  quotients,  12  and  7,  used  in  their  room  ;   lOOxT 
=700,  and  700-T-l2=i.5S-333^,  as  before. 
Rule  2. 

1.  Consider  whether  you  are  to  add  to,  or  subtract  from 
the  given  currency,  in  order  to  find  t]\(i  amount  in  the  cur- 
rency required  When  the  dollar  is  fewer  siiillings  in  the 
given  currency  than  in  the  required,  you  are  to  add ;  when 
it  is  more,  you  are  to  subtract. 

2.  To  find  how  much  you  are  to  add  or  subtract,  fin(\  the 
diff'erence  between  the  value  of  a  dollar  in  the  two  currencie^o 
and  see  what  part  that  is  of  a  dollar  in  the  given  currency. 

3.  Take  such  part  of  the  given  sum,  and  add  it  to,  cr 
subtract  it  from  itself,  as  occasion  requires. 

J^X  AMPLE    1. 

In  Z>100  N.  England  currency,  how  much  Pennsylvania  ? 

1.  The  value   of  a   dollar   in  New-England  currency  i 
fewer  shillings  than  in  Pennsylvania ;  I  am  therefore  to  add 
to  the  given  sum* 

12^ 


102  Exchange. 

Q.  The  value  of  a  dollar  in  New-England  currency  is  Gs., 
and  in  Pennsylvania  7s,  6d.  The  difference  is  Is.  6c?.,  which 
is  I  of  6.-^.  I  am  therefore  to  add'^  of  the  given  sum  to  itself. 
3.  I  find  what  is  ^  of  1.100,  the  given  sum,  and  it  is  L2.j  ; 
and  this  added  to  the  given  sum,  makes  1.125,  which  is  the 
answer. 

Example  2. 

In  JLIOO  New-York  currency,  how  much  New -England  ? 

Operation. 

1.  The  value  of  a  dollar  in  New- York  currency  is  more 
shillings  than  in  New-England  ;  I  am  therefore  to  subtract. 

2.  The  value  of  a  dollar  in  New-York  currency  is  8s.,  and 
in  New-England  6s.  The  difference  is  2:s.,  which  is  -^  of  8s» 
I  am  therefore  to  subtract  i  of  the  given  sum  from  itself. 

3.  I  find  what  is  i  of  LlOO,  the  given  sum,  and  it  is  I/'ES, 
and  this  subtracted  from  the  given  sum,  gives  L75,  for  the 
answer. 

By  the  tw^o  preceding  rules,  the  follow^ing  table  is  con- 
structed. The  parts  to  be  added  or  subtracted,  are  found 
by  the  second  rule,  and  the  multipliers  and  divi&ors  by  the 
first.  It  is  best  to  add  or  subtract,  when  the  numerator  of 
the  fraction  to  be  added  or  subtracted  is  1.  When  that  is 
Bot  the  case,  it  is  best  to  make  use  of  the  multipliers  and 
divisors. 

Tabular  Rules  for  reducing  vayious  Curreneies  to  others. 

Look  for  the  given  currency  in  the  left  hand  column,  and 
then  look  along  the  top  for  the  currency  required,  under 
which,  and  opposite  to  the  given  currency,  is  the  part  of  the 
o-iven  sum  which  is  to  be  added  or  subtracted.  Where 
it  is  more  convenient  to  multiply  and  divi<le,  the  mul- 
tipliers and  divisors  are  put  down,  marked  wiDi  their  proper 
:^iffns. 


Tdhk. 


103 


if 

"^  X 

■§2 
■"X 

5? 

""  X 

^  X 

1*0 

Ci3 

Br.  America,  i     Sterling. 

0   ^ 

S  X 

Cfi 

1^" 

en 

U  00 

^  X 

c« 

5 

c5       . 

-:3 

u 
el 

-  X 

CO    X 

1-  »o 

!i5 

Sx 

'■a 

1^ 

° -I- 

C3    X 

is 

^  x 

J 

fcX) 

c 

•it 

i 

rt    .- 

^  X 

• 

3 

11 

o 

r3 

c3 

c  -I- 
^  X 

0  -I- 

2  CO 

^  X 

0  •!• 

3  ^ 

^  X 

3  ^ 

«  teg 

If 

< 

1 

1  . 

iC4  Exchange. 

Case  3. 
To  reduce  the  weights,  measures,  and  coins  of  one  country, 
to  those  of  another. 

Rule. 
The  proportion  of  the  weight,  measure,  or  value,  of  one 
country,  to  some  known  weight,  measure,  or  value,  of  tlie 
other,  is  usually  stated  in  the  question,  or  is  found  in  tliQ 
tables  of  Reduction  ;  and  then  the  answer  can  be  found  bj 
the  rule  of  three. 

Example  1. 
The  great  bell  of  Moscow  weighs  12500  6  Russian  poods  ; 
how  many  tons  is  that,  2  poods  being  equal  to  Tltb,  avoirdu- 
pois ? 

Operation, 
poods,      lb*  poods* 

2  :  71  :  12500*6  :  the  answer ;  which  pro- 
portion being  worked  out,  gives  443771-3/6.,  or  l98  T.  2  Cwt» 
27'3lb.  for  the  answer. 

Example  2. 
Abraham  gave  400  shekels  of  silver  for  the  cave  and  field 
©f  Machpelah  ;  how  much  is  that  in  federal  money  ? 
Operatimi. 
Ey  the  tables,  I  find  that  the  shekel  of  silver  is  4  drachmae, 
and  that  one  drachma  is  i2yV=^  cents.     400  shekels  is   l600 
drachmas ;  therefore, 

]dr.     :     l2jWcts,     ::     \600dr,     :     the   answer;  which 
proportion  being  worked  out,  gives  D202-77J,  for  the  answer. 
Example  .'. 
The  head  of  Goliah's  spear  weighed  600  shekels  of  iron  ; 
how  many  pounds  is  that,  avoirdupois  weight  ? 
Qpn^dtlon. 
By  the  tables,  I  find  the  shekel  <o  be  4  drachraee,  and  the 
drachma  to  be  '^dwt.  6|^rs.   Troy ;  and  also  that  7000;^^* 
Troy  are  equal  to    lib,  avoirdupois.     eOO  shekels  il  24o0 
drachmse ;  therefore, 

\dr,  :  o4|^r.  ::  ZiOOdr.  :  the  weight  inTroy  .^t^, 
which  proportion  being  v/orkcd  out,  gives  13  400 »*rs.  Troy, 
for  the  weight.     Then, 

7U00^i,''rs.  :  \lh  ::  13 1400 ^rs.  :  the  answer; 
which  proportion  being  w^orked  out,  gives  18/6.  \%ox.  dw 
for  the  answer. 

rXAMPLK    4. 

What  was  the  value  per  bushel,  in  federal  ironcy,  *)f  t!  s 
fine  fiour  mentioned  in  2  Kings  7.  i^>  being  1  seah  for  a 
shekel  of  silver  ? 


Exchange. 


105 


Operation. 
By  the  tables,  I  find  the  seah  to  be  §  of  the  epha,  and  the 
epha  60  pints,  wine  measure,  and  l5  solid  inches.  The  seah, 
therefore,  is  20  wine  pints  and  5  solid  inches,  that  is,  582  J 
solid  inches.  A  shekel  of  silver  is  4  drachmae,  of  \2-^£jCts, 
each,  that  is  50||  cents;  and  a  bushel  is  2150|  solid  inches^ 
Therefore, 

5S2^in.     :     50||cfs.     ::     2l50|in.     :     the  answer ; 
which  proportion  being  worked  out,  gives  Dl^Sr^^f ,  for  the 
answer, 

J\*ote  1.  Sometimes  the  rate  of  exchange  is  stated  at  a  certain  sum  per  cenL 
That  is,  ZlOOin  one  country  are  worth' so  much  more  than  //lOO  in  the  other. 
Wiien  this  is  the  case,  consider  in  which  currency  lAOO  is  worth  the  most. 
If  in  the  currency  required,  add  the  given  rate  per  cent  to  ilOO,  and  make 
it  the  first  term  of  a  proportion  ;  make  XlOO  the  second  term;  and  the  sum 
in  the  given  currency  the  third  ;  and  proceed  as  in  the  rule  of  three. 

JVote  2.  When  the  exchange  is  in  favor  ot  the  given  currency,  make  ZlOO 
the  first  term  ;  7^100  added  to  the  rate  per  cent,  the  second  ;  and  the  given 
currency  the  third. 

Example. 
Philadelphia  is  indebteii  to  London,  |g  1400,  Pennsylvania 
currency ;  how  much   is  that  in   sterling  money,  when  the 
exchange  is  at  64  per  cent  in  favor  of  London  P 
Operation. 
Here,  I  consider,  that  since  the  exchange  is  64  per  cent  in 
favor  of  London,  £  1 00  sterling  is  equal  to  ^164  Pennsylva- 
nia ;  and  so  I  say,  as  =gl64  is  to  ^10;/,  so  is  i61400  to  the 
answer;  which  proportion  being  worked  out,  gives  ^853  .. 
13  ..  2,  and  a  little  more,  for  the  answer^ 

Questions  on  the  foregoing. 


What  is  equation  ? 

How  do  you  find  the  equated  time  ? 

How  do  you  prove  equation  ? 

Is  this   method   of  equation  exactly 

equitable  ? 
Why  so? 
"What  is  barter  ? 
By  \vliat  rnle  do  you  woik  questions 

in  b.'irter  ? 
\\  hut  is  loss  ar.d  »fiin  ? 
How   do  you  work  questions    \%  loss 

lAii]  gain  i' 
Wliut  IS  fellowship? 
"Wliat  IS  the  first  case,  and  rule  ? 
The  second,  and  rule  ? 
What  is  ihe  method  of  proof  ? 
What  is  exchange  I 
What  is  j_)af  ? 


How  do  you  reduce  pounds  to  dol- 
lars ? 

Mow  (loyou  reduce  dolliars  to  pounds? 

What  is  the  shorter  method  of  redu- 

I     V'"r?  p'J'ii'ds  sterling  to  »lollars  ? 

What  is  ihc  rrasoii  of  this  rule  ? 

How  do  you  rcduec  the  currency  of 
one  stule  to  that  of  another,  by  the 
first  method  ? 

How,  1)'  the  "second  method 

How  do  you  reihice  ih«^  weights,  mea- 
sures, and  coins  of  one  country,  to 
tliose  of  another  ? 

When  the  rate  of  exchange  is  a  cer- 
tain sum  percent  aiul  in  favor  of 
the  required  currercy^how  do  you 
proceed  ? 

How,  wh^  n  it  is  in  favor  of  the  jjivea 
fiurj'Ciicy  ? 


106  JBuodecimals, 

DUODECIMALS, 

Are  fractions  so  called  because  thej  decrease  by  twelves^, 
inches  being  twelfths  of  a  foot,  which  is  the  whole  number ; 
and  seconds  being  twelfths  of  an  inch,  tiiirds  twelfths  of  a 
second,  and  so  on.  They  are  chiefly  useful  to  ascertain  the 
superficial  or  solid  content  of  such  things  as  are  measured  by 
feet,  inches,  &c. 

Addition  and  Subtraction  of  Duodecimals  are  performed 
as  in  Compound  Addition  and  Subtraction. 


MULTIPLICATION  OF  DUODECIMALS. 

Rule. 

1.  Place  the  multiplier  under  the  multiplicand,  in  such 
manner  that  the  feet  of  the  multiplier  stand  under  the  lowest 
denomination  of  the  multiplicand. 

2.  Begin  with  the  lowest  term  of  the  multiplier,  and  pro- 
ceed to  the  left,  placing  the  product  of  the  lowest  term  of  the 
multiplicand  under  its  multiplier,  and  so  on  through  all  the 
terms,  carryirg  1  for  every  1^2, 

3.  Take  the  second  term  of  the  multiplier,  and  proceed  in 
the  same  manner  ;  and  so  on  through  all  the  terms  ;  and  the 
sum  of  the  products  will  be  the  answer* 

Example  1. 
Multiply  Sft.  6m.  9  sec.  by  7fU  3in.  8  sec. 
Operation. 
Ft.   in.  sec.  th.  /.  Here,  I  first  place  the  multiplie]? 

8  ..    6  ..  9  so  that  7,  the  feet,  may  stand  under 

9,  the  seconds  of  the  multiplicand, 
and  the  other  denominations  of  the 
multiplier  in  order  towards  the 
right.  I  then  begin  witli  8  and  9, 
the  lowest  denominations  of  the 
multipliei:  and  multiplicand  ;  and 
62  ..  6  ..  7  ..  9  ..  0  Jns.  say,  8x9  is  72,  which  being  6  times 
12,  I  set  down  0  under  the  multi- 
plier &,  and  carry  6  to  the  next.  Then,  I  say,  8x6  is  48, 
and  6  1  carried  is  54,  which  being  4  times  12  and  6  over,  I 
set  down  6  and  carry  4.  Then,  8x8  is  64,  and  4  I  carried 
is  68,  which  is  5  times  12,  and  8  over.  Set  down  8,  and 
carry  5.  But  as  there  is  no  other  term  to  multiply,  I  set 
down  the  5  in  the  next  place. 

I  then  take  3,  the  next  term  of  the  multiplier,  and  say^ 
3x9  i::  27^  which  is  2  time^  12,  and  3  over.     Set  down  3  \ic 


7. 

.3. 

.8 

5  . 

.8. 

.6. 

.0 

o  ^ 

..    I  . 

.8. 

.3 

59, 

..  11  . 

.3 

Duodecimals, 


107 


der  the  multiplier  3,  and  carry  2  to  the  next.  In  like  man- 
ner, I  proceed  till  1  have  multiplied  all  the  terms  of  the  mul- 
tiplicand by  3. 

I  then  take  7,  the  last  term  of  the  multiplier,  and  say  7x9 
is  63,  which  is  5  times  12,  and  3  over.  Set  down  3  under 
the  multiplier  7,  and  carry  5  to  the  next.  And  so  on,  till  I 
have  multiplied  all  the  terms  of  the  multiplicand  by  7.  I 
then  add  up  these  several  products,  and  the  answer  is  6^ft 
6in,  7  sec.  9  thirds. 

Example  2. 
How  many  feet  of  wood  are  there  in  a  load  that  is  8/^  6in* 
long,  2/f.  Sin,  wide,  and  2ft,  din,  high  ? 
Operation, 
ft.  in.  sec.  th, 
8 ..  6  length. 

2  ..  3         w  idth. 


2..  1  ..6 
17..0 


19. 


1  ..  6 

3 ..  9  height. 


14..4..  1  ..6 
57..  4..  6 

71 ..  8  ..  7 ..  6  Ms, 
Questions  on  the  foregoij^g. 


What  are  Duodecimals  ? 

For  what  are  they  useful  ? 

How  are  addition  and  subtraction  of 

duodecimals  performed  ? 
Ih  multiplication  of  duodecimals,  how 

do  you  set  down  the  multiplier  ? 

Exercise  52, 


Where  do  you  begin  to  multiply  ? 
For  how  many  do  you  carry  1  ? 
Where  do  you  set  down  the  product 
of  each  figure  of  the  multiplicand  ? 
What  will  be  the  answer  ? 


Tell  what  is  the  least  common  muiti- 


pie  of 

Am. 

pie  of 

Ans, 

5,  4,  and  5, 

60 

3,  7,  and  6, 

42 

6,  7,  and  8, 

168 

7,  8,  and  S, 

lr,8 

7,  8,  and  9, 

504 

5,  7,  and  8, 

280 

S,  6,  and  7, 

210 

4,  5,  and  7, 

140 

S,  5,  8.  and  10, 

120 

5,  3,  and  9, 

4S 

2,  7,  and  14, 

28 

4,  6,  and  7, 

84 

6,  10,  and  8, 

120 

6,  7,  and  12, 

84 

§,  3,  and  lU 

66 

3,  5,  and  8, 

120 

Tell  what  is  the  least  ©ommon  multi- 


loa 

Exercises. 

Exercise  53. 

Eeduce  to 

a  common  denominator,     | 

Reduce  to 

a  common 

denominator, 

I  and 

I. 

^Ins.  1  and  | 

j  and 

|,    ^ns. 

9 

and  II 

^  and 

1 

'2> 

f  and  f 

fand 

1. 

S.4 
3  0 

'and  If 

i  and  i, 

fandf 

1  and 

h 

l\ 

and  y\ 

1  and 

h 

A  and  T^-j 

*and 

i. 

3  6 

and  j\ 

fahd 

h 

^V  and  If 

fand 

I. 

If 

and^V 

Exercise  54. 

Tell  the  sum 

of 

.4ns. 

1  ell  the  sum  of 

Ans* 

^  and 

i. 

s. 

4 

f  and 

!. 

■*  1  .5 

^  and 

§. 

6 

fand 

1. 

Hf 

i  and 

f. 

H 

f  and 

h 

1* 

i  and 

i 

A 

f  and 

i. 

If 

fand 

f. 

h\         1  and 
Exercise  55. 

h 

U 

From 

take 

v^i*. 

From 

take 

Aii^. 

2. 

JL 

i 

3 

Z 

.1-3 

4> 

^f 

2 

4» 

T> 

28 

f> 

i. 

i 

1. 

#   h 

JL 

I  5 

4> 

i. 

* 

f. 

h 

3V 

h 

h 

A 

1. 

h       • 

tV 

1 

1 

7 

1 

3 

1 

2» 

7» 

T8          '         2» 

Exercise  56. 

7> 

T? 

Multiply 

by 

JltlS. 

Multiply 

by 

v5;?*. 

I 

2"» 

^. 

i 

h 

|. 

f 

!' 

1 

i 

i> 

"t* 

t'o 

t' 

I. 

t\          h 

1.- 

1 

1 

2 

2                      1 

5 

5 

t> 

3> 

15                  -2> 

-6» 

IT 

4 

3 

?              X 

2 

a 

i» 

?» 

5                  3J 

Exercise  57. 

^' 

2T 

MuIiWy 

by 

^?w. 

Multiply 

by 

Atis, 

1, 

1. 

f 

H. 

-J. 

i 

2, 

1. 

J| 

2§. 

*, 

i 

3. 

1. 

2f 

Si. 

il. 

if 

4, 

f. 

2|     1 

4|. 

h 

2* 

5, 

1 

"2> 

2|     1 

5|. 

-h 

4A 

Exercise  58. 

Sivide 

by 

.^ns. 

Diride 

fey 

JtM* 

1. 

^. 

2 

4, 

4. 

H 

i. 

h 

1: 

i. 

i. 

i 

4. 

h 

f 

i. 

h 

f 

}. 

h 

i 

i. 

h 

1 

i. 

h 

H 

i. 

h 

If 

^o'erciseso 

\ 

Exercise  59. 

Tell  the  sum  of 

Ans, 

s. 

s.     d!.          £.    s.      d. 

*. 

s. 

d. 

2, 

..    3 ..    6  and     1  .•    4 ..    3, 

3. 

.    7.. 

9 

3. 

..  15  ..    8  and    2  ..    3  ..    5, 

5. 

.19.. 

1 

2. 

..  10..    1  and    3..  15..  11, 

6. 

.    6.. 

0 

2, 

..13..    6  and    1..  16..    8, 

4. 

.10.. 

2 

10, 

..10..  10  and    2..    2..    2, 

12. 

.13.. 

0 

5. 

.    5 ..    5  and    6 ..    6 ..    6, 

11  . 

.11.. 

11 

13, 

..  13..    Band  14..  14..    4, 

28. 

.    8.. 

0 

5. 

..    6..    7  and    8..    9..  10, 

13, 

.  16  .. 

5 

4, 

,.    5 ..    6  and    5 ..    6 ..    7, 

9. 

.12.. 

I 

3. 

..    5..    4  and    4..    3..    5, 

7. 

.    8.. 

9 

6. 

..    4  .*    3  and    3 ..    4 ..    6, 

9. 

..    8.. 

9 

3. 

,.    7 ..    6  and    4 ..    3 ..    6, 

Exercise  60, 
Tell  the  sum  of 

7. 

..11.. 

0 

s. 

d.         s.    d,         s.    d. 

.€. 

s. 

d. 

3, 

,.  6  and  4  ..,6  and  5  ..  6, 

13.. 

6 

2. 

..  4  and  3  ..  4  and  4  ..  4, 

10.. 

0 

3. 

.  5  and  4  ..  6  and  5  ..  7, 

13.. 

6 

7. 

,.  6  and  8  ..  8  and  9  ..  9, 

1 . 

.    5.. 

11 

3. 

.4  and  5  ..  6  and  7..  8, 

16.. 

6 

8. 

..  7  and  6..  5  and  4..  3, 

19.. 

5 

6. 

.  5  and  6  ..  6  and  6  ..  7, 

19.. 

6 

4. 

,.  S  and  5 ..  6  and  6 ..  4, 

16.. 

6 

3. 

..5  and  3  ..  9  and  3..  11, 

11.. 

1 

il  . 

..  3  and  9 ..  3  and  5  ..  3, 

1. 

.    5.. 

9 

6. 

..  8  and  7.-6  and  8 ..  4, 

1. 

.    2.. 

6 

2, 

..  7  and  6 ..  3  and  4  ..  5, 

Exercise  61. 

13.. 

3 

From                                  take 

Ans. 

€• 

s.      d.             £.      s.        d. 

£' 

S, 

d. 

1.. 

1..    0,                      13..    6, 

7.. 

6 

2.. 

3..    6,                      19..    0, 

1 . 

.    4.. 

6 

18..    5,                      13..    6, 

4.. 

11 

2.. 

2 ..    2,                1  ..    3  ..    5, 

18.. 

9 

1.. 

17..    8,                      18..    9, 

18.. 

11 

3.. 

18..    0,                1..  19..    6, 

1. 

.18.. 

6 

1.. 

12..  10,                      18..    6, 

14.. 

4 

1.. 

1..    1,                     19..    9, 

1.. 

4 

2.. 

2..    2,                1..  11..  11, 

10.. 

3 

3.. 

3..    3,               2..  19..    0, 

4.. 

S 

4.. 

5..    6,               3..    4..    5, 

1. 

.    1.. 

1 

5,. 

4  «•    3j                 3  ••    4  ••    5f 
K 

1. 

.  19 .. 

10 

109 


110 

Exercises^- 

ExERC 

isE  62. 

Tell  what  is 

^?w. 

Tell  what  is 

An3. 

S.      d. 

£.' 

s.    d. 

S,      d. 

£' 

s,  d. 

3  times  6  ..^8, 

1 

.    0..0 

3  times  9  ..  9, 

1 

.    9..  3 

4  times  2 ..  3, 

9..0 

5  times  8  ..  8, 

2, 

.    3..4 

5  times  10..  6, 

2 

.  12..  6 

6  times  2 ..  3, 

13  ..6 

€  times  5  ..  6, 

1 

..  13..  0 

5  times  4 ..  8, 

1  . 

.    3..  4 

7  times  4  ..  6, 

1, 

.  11  ..  6 

9  times  5  ..  5, 

2 

.    8..9 

8  times  8  ..  8, 

3 

.    9..4 

3  times  10..  10, 

1 

..12..6 

7  times  7  ..  7, 

2 

.  13..  1 

8  times  6  ..  7, 

2 

.  12..  8 

4  times  6  ..  6, 

1 

.    6..0 

ExERC 

7  times 9..  10, 

ISE  63. 

3 

.•8..  10 

Tell  what  is  the 

^71S. 

Tell  what  is  the 

Ans. 

£.   s.    d. 

5.      d. 

£.    s.    d. 

S.      d. 

4th  of  1 ..  8..  0, 

7..0 

10th  of          9..  2, 

0..  11 

Sd  of  1 ..  4  ..  6, 

8..  2 

6th  of  1  ..    5  ..  6, 

4..    3 

5tli  of  3  ..  0  ..  0, 

12. .0 

4thof  1..  12  ..8, 

8  ..    2 

6tk  of  1  ..  4  ..  6, 

4..  1 

5th  of  I ..  15  ..  1(J 

, 

7..    2 

7th  of  2  ..  2  ..  0, 

6..0 

7th  of  2  ..    2  ..  7, 

6..    1 

3d  of  I  ..  1 ..  3, 

7..  1 

8th  of  1  ..  14  ..  8, 

4..     4 

8th  of  2  ..  8  ..  0, 

6..0 

3d  of         13  ..  9, 

4..    7 

9th  of  1  ..  7 ..  9, 

3..  1 

ExERC 

4th of        17..  8, 

ISE  64. 

4..    5 

Tell  how  many  times 

Ans. 

Tell  how  many  times 

Am. 

s.    d.       s.     d. 

s.     d,      s.    d. 

1  ..  2  in    4  ..  8, 

4 

3  ..  5  in  10  ..  3, 

3 

2  ..  6  in  12  ..  6, 

5 

1..  3in    6..  3, 

5 

1.1  in    6  ..  6, 

6/ 

2  ..  5  in    9  ..  8, 

4 

2  ..  i  in    6  ..  3, 

3 

1  ..  5  in    8  ..  6, 

6 

1  ..  5  in    5  ..  8, 

4 

1  ..  7in    7  ..11, 

5 

1 ..  2  in    7  ..  0, 

6 

2 ..  9  in    8  ..  3, 

3 

2  ..  2  in    6  ..  6, 

3 

1 ..  1 1  in  7 ..  8, 

4 

3  ..  1  in  12..  4, 

4 

2..  Sin  11  ..3, 

5 

INVOLUTION, 

Is  the  raisitig  of  numbers  to  powers. 

A  power ^  is  a  number  produced  by  multiplying  a  smaller 
number  by  itself  a  certain  number  of  times. 

The  smaller  number  so  multiplied  to  produce  a  power,  is 
called  the  root  of  that  power. 

Thus,  3x3=9.    Here,  9  is  the  power,  and  3  is  its  root. 

4X4X4=64.     Here,  64  is  the  power,  and  4  is  its  root. 

5x5x5x5=625.    Here,  625  is  the  power,  and  5  is  its  root. 


Involution.  Ill 

When  a  number  is  multiplied  by  itself  once,  the  product 
s  called  the  second  power,  or  square,  of  that  number  ;  and 
the  number  multiplied  is  called  the  square  root  of  that  pro- 
duct. As,  3x3=9  5  here,  9  is  the  second  power  or  square 
of  3  ;  and  3  is  the  square  root  of  9.  Again,  5x5=25 ;  here> 
25  is  the  square  of  5,  and  5  is  the  square  root  of  25. 

When  a  number  is  multiplied  by  itt^elf,  and  the  product  so 
produced  is  multiplied  again  by  the  same  number,  the  second 
product  is  called  the  third  power,  or  cube  of  that  number  ; 
and  the  number  is  called  the  cube  root  of  that  second  pro- 
duct. As,  4X4x4=64  ;  here,  64  is  the  third  power  of  4, 
and  4  is  the  cube  root  of  64.  Again,  3x3x3=^27;  here,  27 
is  the  cube  of  3,  and  3  is  t|ie  cube  root  of  £7. 

W' hen  a  number  is  multiplied  by  itself,  and  that  product 
again  by  the  same  number^,  and  the  second  product  again  by 
the  same  number,  the  third  product  is  called  the  fourth  pow- 
er, or  biquadrate  of  that  number  ;  and  the  number  is  called 
the  fourth  root,  or  biquadrate  root  of  that  third  product.  As, 
2x2x2x2  =  16;  here,  16  is  the  fourth  power  of  2,  and  2  is 
the  fourth  root  of  16.  Again,  3x3x3x3=81;  here,  81  is 
the  fourth  power  of  3,  and  3  is  the  fourth  root  of  81. 

Rule. 

To  raise  a  given  number  to  any  given  power,  multiply  that 
number  by  itself,  and  that  product  by  the  given  number,  and 
so  on,   till  the  number  of  multiplications  shall  be  one  less 
than  the  number  of  the  given  power. 
Example. 

Find  the  sixth  power  of  2. 

Operation, 
2 
2 

4  square. 
2 

8  cube. 
2 

16  biquadrate. 
2 

33  5th  power. 
^  2 

64  6th  power*    Ans. 


112 


Evolution* 


A  vulgar  fraction  is  involved  to  any  power  required,  by 
involving  the  numerator  to  that  power,*^  and  then  involving 
the  denomioator  to  the  same  power.  Thus  the  square  of  § 
is  f,  and  the  cube  off  is  -/. 

EVOLUTION, 

Is  extracting  or  finding  the  roots  of  given  powers. 

Numbers,  with  respect  to  their  roots,  are  rational,  or  irra- 
tional. 

A  rational  number  is  one  of  which  the  exact  root  can  be 
found. 

An  irrational  number,  or  surd,  is  a  number  of  which  the 
exact  root  cannot  be  found. 

The  same  number  may  be  rational  with  respect  to  one  of 
its  roots,  and  irrational  with  respect  to  another.  Thus,  4, 
with  respect  to  its  square  root,  is  rational ;  but  with  respect 
to  its  cube  root,  it  is  irrational ;  for  the  square  root  of  4  can 
be  found,  but  the  cube  root  of  4  cannot  be  found  exactly. 

Table  of  Powers  and  Roots. 


Roots, 

2 

3 

4 

5 

6 

7 

8 

9 

Squares, 

I  4 
I  8 

9 
27 

16 

25 

36 

49 

64 

81 

Cubes,  1 

64 

125 

216 

343 

512 

7&9 

4th  pow.  1 

16 

81 

256 
1024 

625 
3125 

1296 

2401 

4096 

6561 

5th  pow.  ] 

32243 

7776 

16807 

32768 

59049 

6th  pow.  ] 

I  64 

739 

4096 

15625 

46656 

117649 

262144 

531441 

To  extract  the  Square  Root. 
Rule. 

1.  Distinguish  the  given  number  into  periods  of  two 
figures  each,  beginning  at  the  place  of  units,  and  marking 
every  second  figure  from  the  place  of  units  to  the  left, 
in  whole  numbers,  and  from  the  decimal  point  to  the 
right  in  decimals,  if  any,  annexing  a  cypher  to  the  decimals, 
if  necessary  to  make  an  even  number  of  places. 

2.  Begin  with  the  left  hand  period,  and  find  by  trial  the 
greatest  square  it  contains,  and  set  its  root  on  the  right  hand 
of  the  given  number,  separated  from  it  by  a  curve  line,  for 
the  first  figure  of  the  required  root. 


Evolution,  1 1 3 

* 
5.  Subtract  the  square  thus  found  from  the  said  period, 
and  to  the  remainder  bring  down  and  annex  the  next  period, 
for  a  dividend. 

4.  Double  the  root  already  found,  for  a  defective  divisor. 

5.  Find  how  often  the  defective  divisor  is  contained  in  the 
dividend,  exclusive  of  its  right  hand  figure;  and  the  figure 
denoting  that  number  of  times,  will  be  the  next  figure  of  the 
root,  probably, 

6.  Complete  the  divisor,  by  annexing  at  the  right  hand  of 
it  the  last  figure  of  the  root. 

7.  Multiply  the  divisor  so  completed  by  the  last  figure  of 
the  root,  and  subtract  the  product  from  the  dividend. 

8.  Bring  down  another  period,  and  find  another  figure  of 
the  root  in  the  same  manner ;  and  so  on,  through  all  the  pe- 
riods. 

JS'ote  1.  The  reason  for  dividing  the  given  number  into  periods  of  two 
figures  each,  is,  that  the  square  of  any  one  figure  is  never  more  than  two 
figures.*  Heiice^  there  will  be  as  many  places  of  whole  numbers  in  the  root, 
as  there  were  periods  of  whole  numbers  in  the  square. 

JVote  2m  It  will  sometimes  happen,  that  on  multiplying  the  complete  divi- 
so^  by  tliejast  figure  of  the  root,  the  product  will  be  greater  ithan  the  divi- 
dend. In  that  case,  you  mwst  try  the  next  lower  figure,  and  if  that  prove 
too  great,  the  next,  ami  so  on. 

J^foteS,  When  you  have  proceeded  through  all  the  periods  of  the  given 
number,  and  there  is  a  remainder,  the  operation  may  b^*  continued  further, 
if  required,  and  another  figure  of  the  root  found,  by  annexing  two  cyphers  to 
that  remainder,  and  proceeding  as  before.  If  thai  remainder  consisted  of 
whole  numbers,  the  first  figure  found  by  annexing  a  period  of  cyphers,  will 
be  the  first  decimal,  and  so  on. 

JYote^.  The  square  root  of  a  vulgar  fraction  may  be  found  by  reducing  it 
to  its  lowest  terms,  and  extracting  the  root  of  each  term. 

Or,  the  numerator  and  denominator  of  the  given  fraction  may  be  multiplied 
together,  and  the  square  root  of  that  product,  being  extracted,  may  be  made 
the  numerator  to  the  denominator  of  the  given  fraction,  or  tlie  denominator 
to  the  numerator  of  it,  for  the  answer 

But  if  the  exact  root  of  it  cannot  be  found  by  either  of  these  methods,  an 
approximation  to  it  may  be  found,  by  reducing  the  vulgar  fraction  to  a  deci- 
mal  and  extracting  its  root  to  as  many  places  as  shall  be  thought  necessary, 

JVote  5  A  mixed  number  in  vulgar  fractions  must  be  reduced  to  a  mixed 
liumber  in  decimals,  and  the  root  extracted  as  before. 

Proof. 

Squire  the  root,  when  found,  and  add  in  the  remainder,  if 
any  ;  and  the  sum  will  be  equal  to  the  given  number,  ii  the 
work  is  right. 

Example. 

What  is  the  square  root  of  552*25  f 

K2 


U4  SiSoluiioni 

«  >■ 

Operation.  Explanation, 

...  1.  I  distinguish  the  given  numbei' 

552'25(2S'5  Ans.    into  periods  of  two  figures  each,  be- 
4  ginning  at  the   place  of  units,    and 

—  marking  ever  j  second  figure  each  way  5 

43)152  and  have  three  periods,  the  first  con- 

129  sisting  of  5,  the  second  of  52,  and  the 

third  of -25. 

465)2325  2.  I  take  the  left  hand  period,  which 

2325  is  5 J  and  try  how  great  a  square  num- 

ber  I  can  find  in  it,  which  is  4,  the  root 
of  which  is  2.  So  I  set  down  2  far  the  first  figure  of  the  re- 
quired root. 

3.  I  set  down  4,  the  square  thus  found,  under  5,  the  first 
period,  and  subtract;  and  the  remainder  is  !.  To  this  re- 
mainder, I  bring  down  and  annex  the  next  period,  52 ;  and 
I  have  1 52  for  a  dividend. 

4.  T  double  2,  the  root  already  found,  and  it  makes  4,  Which 
I  set  down  at  the  left  of  the  dividend,  for  a  defective  divisor. 

5.  I  seek  how  often  4,  the  defective  divisor,  is  contained  in 
15,  the  dividend  with  the  exception  of  the  right  hand  figure ; 
4  in  15  is  3  times.  So  I  set  down  3  for  the  second  figure  of 
the  root. 

6.  I  now  complete  the  divisor,  by  annexing  at  the  right 
hand  of  4,  the  defective  divisor,  5,  the  second  figure  of  the 
root ;  and  it  makes  43. 

7.  1  multiply  43,  the  divisor  so  completed,  by  3,  the  secotid 
figure  of  the  root;  and  it  makes  129,  which  1  subtract  from 
the  dividend,  152  ;  and  the  remainder  is  23. 

8.  To  this  remainder,  ^3,  1  bring  down  and  aiinex  the 
next  period,  25  ;  and  it  makes  2325,  for  a  dividend. 

9.  I  double  the  root  already  found,  23,  for  a  defective  di- 
visor ;  and  it  makes  46. 

10.  I  seek  how  ofteii  46  is  contained  in  232,  which  is  the 
dividend  excepting  its  right  hand  figure  ;  and  find  it  5  times. 
So  I  set  down  5  for  the  third  figure  of  the  root. 

11.  I  now  complete  the  divisor,  by  annexing  at  the  right 
hand  of  46,  the  defective  divisor,  5,  the  third  figure  of  the 
root ;  and  it  makes  465. 

12.  I  multiply  465,  the  divisor  go  completed,  by  5,  the 
l-hird  figure  of  the  root ;  and  it  makes  2325,  which  being  sub- 
iFacted  from  the  dividend,  there  is  no  remainder. 


Evolution*  115 

The  figures  of  the  root,  therefore,  are  235.  But  as  I  had 
in  the  given  number  only  two  periods  of  whole  numbers,  I 
must  kave  only  two  whole  numbers  in  the  root.  So  I  insert 
a  decimal  point  between  the  second  and  third  figures,  and  the 
answer  is  23*5. 

To  prove  the  operation,  I  square  the  root  so  found  ;  that 
is,  I  multiply  23*5  by  23-5,  and  it  makes  552'QS ;  which  be- 
ing equal  to  the  given  number,  I  conclude  the  work  is  right. 

JSfote  6.  The  areas  of  similar  figures  ere  to  each  other  as  the  squares  of 
their  similar  dimensions. 

^^K  Example. 

'^^  If  a  certain  field,  one  side  of  which  measures  50  rods,  con- 
tains 6  acres,  how  much  does  another  field  contain,  of  the 
same  shape,  the  similar  side  of  which  measures  50  rods  ? 

Operation, 


30 

50 

30 

50 

900 

:  2500  : 
6 

:  6  : 

900)15000(16- 
900 

66+  Ans 

6000 

5400 

6000 

5400 

6000 

5400 

600 

J\'ote  7.  The  square  of  the  longest  side  of  a  right  angled  triangle  is  equal 
ta  the  sum  of  the  squares  of  the  other  two  sides. 

Example. 

The  height  of  a  certain  wall  is  17  feet,  and  there  is  a  ditch 
at  the  foot  of  it  20  feet  wide ;  how  long  must  a  ladder  be,  to 
Teach  from  the  outside  of  the  ditch  to  the  top  of  the  wall  ? 


116  Evolution. 


Operation. 
height  of  wall.  20  width  of  ditcfi* 


17  20 

1 1 9  400  square  of  width. 

17  289  square  of  height. 

289  square  of  height.  689  sum,  and  square  of  the 

length  of  the  ladder ;  the  square  root  of  which  being  extract- 
ed, is  26-24-1-  feet,  which  is  the  answer. 

Problem. 

The  sum  and  product  of  two  numbers  being  given,  to  find 
the  numbers. 

Rule. 

From  the  square  of  the  sum,  subtract  four  times  the  pro- 
duct, and  the  square  root  of  the  remainder  will  be  the  diffe- 
rence of  the  numbers. 

Example. 

A  and  B  make  up  1000  dollars,  and  trade  till  they  have 
gained  two  hundred  per  cent,  which  gain  is  to  be  divided 
in  proportion  to  the  snare  of  each  in  the  capital  stock.  A's 
gain  is  the  most,  and  is  such  a  sum,  that  if  multiplied  by  B's, 
the  product  would  be  960000  dollars.  What  was  each  man's 
share  of  the  capital  stock  ? 

Operation. 

Here,  the  gain,  being  200  per  cent  on  the  capital  stock,  is 
2000  dollars  ;  and  the  product  of  the  two  parts  is  960000  dol- 
lars.    Therefore, 

2000  960000 

2000  4 


4000000  is  the  square  of  the  sum.  3840000 

3840000  is  4  times  the  product. 

•>.«.>  -.  -.  - 

160000(400  is  the  difference, 
16 

0000 
Having  found  the  difference,  I  add  half  the  difference  t© 
half  the  sum  for  the  greater,  and  subtract  for  the  less  ;  and 
each  person's  share  of  the  gain  is,  A's,  12()0  dollars,  and  B's, 
800  dollars  ;  consequently  their  shares  of  Stock  were^  A'e^ 
600  dollars,  and  B'g,  400  dollars. 


Evolution. 


IIT 


Questions  on  the  foregoing. 


What  is  involution  f 
What  is  a  power  ? 
What  is  a  root  ? 

What  is  a  square  ?  The  square  root  ? 
What  is  a  cube  ?  The  cube  root  ? 
What  is  a  biquadrate  ?  The  biquad- 
ratic root  ? 
What  is  the  rule  for  finding  any  given 

power  ? 
How  is  a  \-u!gar  fraction  involved  ? 
What  is  evolution  ? 
What  is  a  ratid' al  number  ? 
What  is  a  surd  ? 
In  extracting  the  square  root,  what  is 

the  first  thing  to  be  done  ? 
Where  do  you  begin  to  point  off? 
How   many  figures  do  you  put  in  a 

period,  and  why  ? 
How  many  whole  numbers  will  there 

be  in  the  root  ? 
How  do  you  find  the  first  figure  of  the 

'root  ? 
Of  what  does  the  first  dividend  con- 
^    sist  ? 
Of  what  does  the  defective  divisor 

consist  ? 
When  you  have  found  tlfc  defective 

divisor,  how  do  you  find  another 

figure  of  the  root  ? 


How  do  you  complete  the  divisor  ? 

By  what  do  you  multiply  ? 

What  do  you  do  next  ? 

What  if  the  product  to  be  subtract- 
ed is  larger  tiian  the  dividend, 
from  which  it  is  to  be  subtracted  ? 

How  do  you  form  another  dividend  ? 

When  you  have  extracted  the  root 
of  the  given  number,  and  there  is 
a  remaiT.dfc:r,  how  can  you  extend 
the  operation  fuither  ? 

What  isthi^  first  method  of  extracting 
the  square  root  of  a  vulgar  fractiouj 

The  second  ?  The  tliird  ? 

How  do  you  extract  tlie  square  root 
<)f  a  m'xed  number  in  vulgar  frac- 
tions f 

How  do  you  prove  the  operation  ? 

How  do  you  know  the  work  is  right  ? 

What  proportion  have  the  areas  of 
similar  figuies  to  each  other  ? 

What  proportion  have  the  sides  of  a 
right  angled  triangle  to  each  other? 

When  the  sum  and  product  of  two 
numbers  are  given,  how  do  you 
find  '.  he  diflTerence  ? 

When  you  have  found  the  difference^ 
how  do  you  find  the  numbers  ? 


To  extract  the  Cube  Moot. 
Rule. 

1.  Distinguish  the  given  number  into  periods  of  three 
figures  each,  beginning  at  the  place  of  units,  and  marking 
every  third  figure  to  the  left  in  whole  numbers,  and  to  the 
right  in  decimals,  if  any,  adding  cyphers  to  the  decimals,  if 
necessary,  to  make  out  the  last  period. 

2.  Begin  with  the  left  hand  period,  and  find  by  trial  the 
greatest  cube  it  contains,  and  set  its  root  on  the  right  hand  of 
the  given  number,  for  the  first  figure  of  the  required  root. 

3.  Subtract  the  cube  thus  found  from  the  said  period,  and 
to  the  remainder  bring  down  and  annex  the  next  period,  for 
a  dividend. 

4.  Take  three  times  the  sti[uare  of  the  root  already  found, 
for  a  divisor. 

5.  Divide  the  dividend  by  the  divisor,  so  far  as  to  find  one 
quotient  figure,  which  will  be  the  second  figure  of  the  root^ 
pvobablif. 


118  Evolution. 

6.  Subtract  the  cube  of  th^se  two  figures  of  the  root  from 
the  two  left  hand  periods  of  the  given  number,  and  to  the 
remainder  bring  down  and  annex  the  third  period,  for  a  se» 
cond  dividend. 

7.  Find  another  divisor,  and  another  figure  of  the  root,  in 
the  same  manner,  and  so  on,  always  subtracting  the  cube  of 
the  root  found  from  as  many  of  the  left  hand  periods  of  the 
given  number,  as  you  have  found  figures  of  the  root. 

J\''ote  h  The  reason  for  dividing  the  given  liumber  into  periods  of  tlA-ee 
figures  each,  is',  that  the  cabe  ^f  any  one  figur*'  is  never  more  than  three 
figures.  Hence,  there  will  he  as  many  pbces  of  whole  numbers  in  the  root, 
as  there  were  y)eriods  of  whole  numbers  in  the  given  number. 

JSi^oie  2,  It  Aviil  sometimes  happen,  that  tin-  cube  of  the  figures  placed  in 
the  root  will  be  found  greater  than  the  pej  iods  from  whicli  it  h  to  be  sub- 
tracted. In  that  case,  the  last  number  placed  in  the  root  is  too  large,  and 
must  be  made  smaller. 

JVote  3.  When  you  have  prrceeded  through  all  the  periods  of  the  given 
number,  and  there  is  a  remainder,  the  operntion  may  be  continued  further, 
if  required  by  annexing  three  cyphers  to  that  remainder,  and  proceeding  as 
before. 

JVote i  The  cube  root  of  a  Tulgar  faction  may  be  found  by  reducing  it  to 
its  lowest  terms,  and  extracting  the  root  of  each  term.  But  if  the  exact 
root  of  each  term  cannot  be  found,  an  approximation  may  be  made  towards 
it,  by  reducing  the  fraction  to  a  decimal,  and  extrf^cting  its  root,  to  as  many 
places  as  shall  be  requisite. 

A^ste  5  The  cube  root  of  a  mixed  number  in  ^ulgar  fractions,  may  be 
found,  by  reducing  it  to  a  mixed  number  in  decimals,  and  then  extracting  the 
root. 

Proof.  Cube  the  root,  when  found,  and  add  in  the  re- 
mainder, if  any ;  and  the  sum  will  be  equal  to  the  given 
number,  if  the  work  is  right. 

Example.  What  is  the  cube  root  of  34328-12')  ? 

Operation,  Eocplanation* 

1.  I    di>tinguish    the    given 

S4328'125(32'5  Ans,   number   into  periods  of  three 

27  figures  each,  beginning  at  the 

—  place   of  units,    and    marking 

27) 7 "28  first  dividend,  every  third   figure  each  way  ; 

and  have  three  periods,  the  first 

S2768  cube  of  32.        consisting  of  34,   the  second  of 

328,  and  rhe  third  of  •  i  25. 

3072)1560125  second  dividend. 

2.  I  take  the  left  hand  period, 

34328 125  cube  of  525.  which  is  3^,  and  try  h(*w  great 
a  cube  I  can  find  in  it,  which 
IS  27,  the  root  of  which  is  3  ;  so  1  set  down  3  for  the  first 
figure  of  the  required  root 


Evolution.  1  i  9 

0.  I  set  down  S7,  the  cube  thus  found,  under  34,  the  first 
period,  and  subtract,  and  the  remainder  is  7.  To  this  re- 
mainder, 1  bring  down  and  annex  the  next  period,  328,  and  I 
have  7328  for  a  dividend. 

4.  1  square  3,  the  root  already  found,  and  it  makes  9 ;  and 
multiply  that  by  3,  and  it  makes  27,  which  I  set  down  at  the 
left  hand  of  the  dividend,  for  a  divisor. 

5.  I  seek  how  often  27,  the  divisor,  is  contained  in  7328, 
the  dividend,  and  find  the  first  quotient  figure  to  be  2.  So 
I  set  down  2  for  the  next  figure  of  the  root. 

6.  [  cube  32,  the  root  already  found,  and  it  makes  32768, 
which  I  subtract  from  34328,  the  two  left  hand  periods  of  the 
given  number,  and  the  remainder  is  1560,  to  which  I  bring 
down  and  annex  125,  the  third  period,  and  it  makes  1560125 
for  the  second  dividend. 

7.  I  square  32,  the  root  already  found,  and  it  makes  1024 ; 
and  multiply  that  by  3,  and  it  makes  3072,  for  the  second 
divisor. 

8.  1  seek  how  often  3072,  the  second  divisor,  is  contained 
in  1560125,  the  second  dividend,  and  find  the  first  quotient 
figure  to  be  5.  So  I  set  down  5  for  the  third  figure  of  the 
root. 

9.  I  cube  325,  the  root  found,  and  it  makes  34328125, 
which  I  subtract  from  34328125,  the  given  number,  and  no- 
thing remains.  So  that  325  are  the  figures  of  the  root.  But 
because  there  were  two  periods  of  whole  numbers,  and  one 
of  decimals,  in  the  given  number,  there  must  be  two  whole 
numbers  and  one  decimal  in  the  answer ;  and  I  place  the 
decimal  point  accordingly,  and  the  answer  is  32-5. 

To  extract  any  root. 
The  rule  for  extracting  the  cube  root  will  serve  for  ex- 
tracting an^  root,  with  a  little  variation,  as  follows  : 

1.  Distinguish  the  given  number  into  periods  of  as  many 
figures  each,  as  is  the  root  to  be  extracted  ;  that  is,  for  the 
fourth  root,  into  periods  of  four  figures  each  ;  for  the  fifth, 
five,  &c. 

2.  Begin  with  the  left  hand  period,  and  find  by  trial  the 
greatest  power  of  the  same  name  with  the  required  root,  that 
is,  the  fourth  power  for  the  fourth  root,  the  fifth  power  for 
the  fifth  root,  &c.  and  set  the  root  of  that  po%ver  on  the  right 
hand  of  the  given  number,  for  the  first  figure  of  the  root. 

3.  Subtract  the  power  thus  found  from  the  said  period,  and 
to  the  remainder  bring  down  and  annex  the  next  period  for 
a  dividend. 


120  Evolution. 

4.  If  the  required  root  is  the  fourth,  take  four  times  the 
cube  of  the.root  already  found,  for  a  divisor ;  if  it  is  the  fifth, 
take  five  times  the  fourth  power ;  if  the  sixth,  take  six  times 
the  fifth  power,  &c. 

5.  Divide  the  dividend  by  the  divisor,  so  far  as  to  find  one 
quotient  figure,  which  will  be  the  next  figure  of  the  root, 
probably. 

6.  Raise  these  two  figures  of  the  root  to  the  power  which 
is  of  the  same  name  with  the  required  root,  and  subtract  the 
said  power  from  the  two  left  hand  periods  of  the  given  num- 
ber, and  to  the  remainder  bring  down  and  annex  the  third 
period,  for  a  second  dividend. 

7.  Find  another  divisor,  and  another  figure  of  the  root, 
in  the^  same  manner ;  and  so  on,  always  subtracting  the 
power  found  from  as  many  of  the  left  hand  periods  of  the 
given  number,  as  you  have  found  figures  of  the  root. 

Example  1.  What  is  the  5th  root  of  51 53632  ? 

Operatioti. 

5153632(22  Ms. 
32 

80)1953632  first  dividend. 

5153632  5th  power  of  22. 

Example  2.  What  is  the  fourth  root  of  2215534565= 

Operation. 
•        •         • 

221533456(122  ^m. 
1 

4)12153  first  dividend. 

20736  4th  power  of  12. 


6912)14173456  second  dividend. 

221533456  4th  power  of  122* 

Mte,  The  olMervationB  in  the  notes  under  the  rule  for  extracting  the 
cube  root,  will  apply  to  this  rule  for  extracting  any  root,  with  vEriatioB» 
similar  to  those  in  the  rulet. 


(luestions. 
Questions  on  the  foregoinj&. 


l^H 


■ 


lift  extracting  the  cube  root,  what  is 

the  first  thine  to  be  done  ? 
Where  do  yon'oegin  to  point  off? 
How  many  figures  do  you  put  in  a 

period,  and  why  ? 
How  many  whole  numbers  will  there 

be  in  the  root  ? 
How  do  you  find  the  first  figure  of 

the  root  ? 
Of  what  does  the  first  dividend  con- 
sist ? 
Of  what  does  the  divisor  consist  ? 
When   you  have  found  the  divisor, 

how  do  you  find  another  figure  of 

the  root  ? 
How  do  you  form  a  second  dividend  ? 
How  do  you  find  a  third  figure  of  the 

root  ? 
From  how   many  periods  of  the  ,e;i- 

ven   number,  .must    you    always 

ExurisE  65 


subtract  ? 

What  if  the  cube  of  the  figures  pi  a*, 
ced  in  the  root  is  .greater  than  the 
periods  from  which  it  is  to  be  sub- 
tracted ? 

When  you  have  extracted  the  cube 
root  of  the  given  number,  and 
there  is  a  remainder,  how  can  you 
extend  the  operation  further  ? 

What  is  the  first  method  of  extract- 
ing the  cube  root  of  a  vulgar  frac  • 
tion  ? 

What  is  the  second  method  ? 

How  do  you  extract  the  cube  root  of 
a  mixed  number  in  vulgar  frac- 
tions ? 

How  do  you  prove  the  operation  ? 

How  do  you  know  the  work  is  right' 

What  is  the  rule  for  extracting  any 
root  ? 


Tell  what  is  the  ^iis. 

Square  of  3,  9 

Cube  of  2,  8 

Square  root  of  16,  4 

Cube  root  of  27,  3 

Square  of  9,  81 

Cube  of  4,  64 

Cube  root  of  8,  2 

Sq  uare  root  of  64,  8 

Cube  of  5,  125 

Square  of  11,  121 

Cube  root  of  64,  4 

Square  root  of  25,  5 

Square  root  of  144,  12 
Square  of  the  suna  of  2  and  3,    25 

Square  of  the  square  of  3,  8 1 


Tell  what  is  the  ^%is. 

Sura  of  the  squares  of  5  and  6,  61 
Cube  of  the  square  of  2, 
Sum  of  the  cubes  of  2  and  3, 
Difierence  of  the  square  and 

cube  of  2, 
Difierence  of  the  square  and 

cube  of  3, 
Sum  of  the  square  &■  cube  of  2, 
Sum  of  3  and  the  square  of  3, 
Square  root  of  the  cube  of  the 

square  of  2, 
Pro(hict  of  the  square   and 

cube  of  2, 
Quotient  of  the  cube  of  2  by 

the  square  of  2, 


64 
35 


18 
12 
12 


8 


32 
2 


Exercise  66. 
Tell  what  is  the  JlnJi. 

Square  of  the  sum  of  4  and  5,  81 

Sum  of  the  squares  of  4  and  5,  41 

Difference  of  the  cubes  of  2  and  3,  19 

Sum  of  the  cubes  of  2  and  4,  72 

Sum  of  the  square  roots  of  64  and  100,  18 

Square  root  of  the  sum  of  the  square  roots  of  36  &  ?  00,        4 

Square  of  the  difierence  of  tHe  square  roots  of  64  &  25,        9 

Cube  of  the  difference  of  the  square  roots  of  49  and  l6,      2T 

Add  1  to  5x7,  and  tell  the  square  root,  6 

Square  root  of  the  sum  of  the  squares  of  3  and  4,  5 

Sum  of  the  square  of  8  and  cube  of  2,  72 

L 


12!^  £jcercise3. 

Exercise  67. 
Add  4  to  5x9,  and  tell  the  square  root,  Jlns.  7. 

Add  1  to  7x9,  and  tell  the  square  root,  8 

Subtract  20  from  7x8,  and  tell  the  square  root,  6 

Add  2  to  the  square  of  5,  and  tell  the  cube  root,  3 

Subtract  8  from  the  cube  of  4,  and  tell  the  seventh^  8 

Subtract  5  from  6x9,  and  tell  the  square  root,  7 

Add  5  to  the  square  of  5,  and  tell  the  tenth,  3 

Subtract  4  from  the  square  of  5,  and  tell  the  cube  of  1  seventh,  27 
Add  2  to  the  cube  root  of  8,  and  tell  the  difference  between  its 

square  and  cube,  48 
Subtract   1  from  the  square  root  of  4,   and  tell  the  sura  of  its 

square  and  cube,  2 

Tell  5  times  the  sum  of  the  squares  of  3  and  4,  125 

One  third  of  the  sum  of  the  cubes  of  2  and  4,  24 

Half  the  difference  of  the  cubes  of  3  and  2,  9| 
Subtract  the  square  of  5  from  the  cube  of  5,  and  tell  the  square 

root,  10 
Subtract  the  square  of  4  from  the  cube  of  4,  add  1,  and  tell  the 

square  root,  7 
Add  the  square  of  3  to  the  square  of  4,  subtract  5,  £c  tell  the  4tb,      5 

Tell  one  half  of  one  tenth  of  the  square  of  10,  ^^ 

Onehalf  oftwo  thir^Ts  ofthe  square  of  6,  12 

Two  thirds  of  one  half  of  the  square  of  12,  48 

One  half  of  two  thirds  of  three  fourths  of  the  square  of  10,  25 
Add  the  odd  numbers  below  10,  and  tell  the  cube  of  the  square 

root,  125 
Exercise  68. 
Add  3  and  4  and  5  to  5  times  5,                                               Jns.  37 

Take  9  and  8  and  7  from  6  times  6,  12 

Multiply  2  and  3  and  4  by  3  times  3,  81 

Divide  9  and  5  and  10  and  20  by  11,  4 

Add  the  sum  of  6  and  11  to  their  difference,  22 

From  the  sum  of  8  and  13,  take  twice  their  difference,  11 

Multiply  the  sum  of  8  and  14  by  half  their  difference,  66 

Divide  the  sum  of  22  and  14  by  half  their  difference,  9 

Add  the  sum  of  the  squares  of  2  and  3  to  twice  their  difference,  23 
From  the  sum  of  the  squares  of  3  &•  4,  take  twice  their  difference,  11 
Multiply  the  sum  of  the  squares  of  4  and  f>,  by  one  third  of  their 

difference,  123 
Divide  the  sura  of  the  square  and  cube  of  3  by  4,  and  tell  the 

square  root,  3 
Subtract  4  from  the  product  of  the  square  roots  of  49  and  16, 

and  tell  the  square  of  one  half  of  it,  144 
Subtract  the  square  of  4  irom  the  square  of  5,  and  tell  the  square 

of  it,  81 

Multiply  1  and  2  and  3  and  4  and  5,  by  one  fifth  of  it,  45  ^ 
Add  the  square  of  1  to  the  cube  of  1,  and  tell  the  square  of  one 

oneth,  4 


Exercises, 


ISS 


Exercise  69. 


I 


Tell  the 

../f/zs. 

TeU  tiie  difference  of  the 

squares  of 

square  off, 

4 
9 

i  and  J, 

Ans.  3^ 

cube  of  J, 

1 
8 

i  and  :^, 

1 6 

square  of  i, 

t\ 

}  and  J, 

tIt 

cube  of  i, 

oV 

Product  of  tlie 

squares  of 

square  of  |, 

if 

i  and  1, 
J  and  i. 

jh 

cube  of -|, 

H 

Too 

3um  of  the  squares  of 

i  and  ^-, 

i  and  i, 

5 

T6 

i  and  -], 

4^0 

i  and  -J, 

A3 

iandi, 

T-L 

i  and  |, 

o  9 
4  0  0 

1  and  1, 

2  a;-, 

Tell  ^vhat  is  the  sum  of 

2  thirds  af  18  '3c  3  ib^.irths  of  16,  24 

2  fifths  of  35  ik.  5  sixths  of  48,    54 

3  fourths  of  40  6c2thh'dsof  24,  46 

4  fil'ciJS  of  35  Sc  3  fit*ihs  of  ,40,      52 

2  fifths  of  15  &•  3  fiitbs  of  30,       24 

3  sevenths  of  21  &  2  thirds  of  27, 27 

Exercise 
From  take 

2  thirds  of  24, 

3  fourths  of  32, 

2  fifths  of  55, 

3  fifths  of  55, 
3  thirds  of  33, 
2  fifths  of  60, 

2  ninths  of  36, 

3  tenths  of  80, 
2  sevenths  of  28, 
.3  fifths  of  100, 
2  sevenths  of  140, 


Exercise  70. 

Jiii.'i.  Tell  what  is  the  sum  of  Ans. 

2  iff  hs  of  25  &  1  third  of  33,      21 

3  iburthb  of  28  &c  2  thirds  of  18,  33 
3  eighths  of  32  6c  2  fifths  of  15,  18 
3  sevenths  of35^  2  fifths  of  40,  31 

2  fifths  of  25  6c  2  eighth,  of  96,  34 

3  fifths  of  30  6c  2  sevenths  of  77,  40 


n. 


ISfuItipiy 

2  thirds  of  9, 

3  fifths  of  15, 

4  fifths  of  25, 

2  sevenths  of  21, 

3  fifths  of  25, 

2  thirds  of  1 2, 

3  fourths  of  8, 

2  fifths  of  20, 

5  sixths  of  30, 

3  sevenths  of  28, 


3  fourths  of  12, 
2  fifths  of  15, 
2  thirds  of  12^ 
2  sevenths  of  28, 

2  ninths  of  27, 

3  sevenths  of  21, 
2  fifths  of  15, 
2  ninths  of  45, 
2  ninths  oi2T, 

4  fifths  of  60, 
S  eighths  of  96, 
Exercise  72 

by 

3  fourths  of  12, 
2  thirds  of  18, 
1  tenth  of  30, 

1  twelfth  of  21, 

2  thirds  of  9, 

3  fifths  of  15, 
2  thirds  of  1 2, 
2fifthsof  15, 

2  sevenths  of  21, 
2  ^nthscfsr; 


7 
18 

6 
25 
16 
15 

2 
14 

2 
12 

4 


54, 

108 

60 

J2 

90 

48 

4S 

150 


Exercises. 


Exercise  73. 

Divide 

by 

,lfi^. 

2  thirds  of  36, 

2  sevenths  of  21 

, 

4 

3  fourths  of  24. 

2  ninths  of  27, 

3 

4  fifths  of  SO, 

3  fourths  of  16, 

2 

5  sixths  of  18, 

1  ninth  of  45, 

3 

2  fifths  of  60, 

3  eighths  of  l6. 

4 

3  fourths  of  48, 

3  sixths  of  1 8, 

4 

4  fifths  of  60, 

2  ninths  of  27, 

8 

3  sevenths  of  42. 

,         3  eighths  of  1 6, 

3 

2  thirds  of  72, 

2  sevenths  of  42. 

t 

4, 

3  fourths  of  48, 

3  eighths  of  24, 
Exercise  74. 

4 

Divide  the  sum  of 

by 

J?i.?, 

2  thirds  of  24  and  1  half  of  18, 

5, 

5 

2  thirds  of  50  and  3  fourths  of  32, 

4. 

11 

1  fourth  of  48  and  2  thirds  of  24, 

7, 

4 

3  fourths  of  16  and  2  fifths  of  60, 

6. 

6 

4  fifths  of  20  and  2  fifths  of  30, 

7. 

4 

2  sevenths  of  28  &  3  sevenths  of  1 4, 

2, 

7 

3  eighths  of  40  and  3  sevenths  of  35, 

6, 

5 

2  thirds  of  36  and  3  fourths  of  24, 

7. 

6 

Exercise  75^ 

From 

take                     and  divide  by 

Alts. 

2  thirds  of  36, 

1  fifth  of  40, 

4. 

4 

2  thirds  of  24, 

1  fourth  of  24, 

5. 

2 

2  fifths  of  45, 

2  ninths  of  27, 

4, 

3 

3  fourths  of  44, 

3  sevenths  of  21, 

s. 

8 

3  eighths  of  64, 

3  eighths  of  24, 

5/ 

3 

2  ninths  of  108, 

1  ninth  of  36, 

4, 

5 

3  sevenths  of  77, 

I  seventh  of  35, 

4, 

7 

4  fifths  of  95, 

4  ninths  of  45, 
Exercise  76. 

7. 

8 

From 

take               and  mu 

iltiply  by 

Ans. 

2  thirds  of  45, 

3  fifths  of  20, 

3, 

54 

3  fourths  of  16, 

5  sixths  of  12, 

9, 

18 

6  sevenths  of  42, 

7  eighths  of  32, 

5, 

40 

5  sixths  of  36, 

6  sevenths  of  28, 

9, 

54 

3  fifths  of  55, 

4  filths  of  35, 

5, 

25 

8  tenths  of  50, 

3  tenths  of  60, 

3, 

66 

6  sevenths  of  28, 

5  ninths  of  36, 

7,  ■ 

28 

7  eighths  of  48, 

6  sevenths  of  42, 

9, 

54 

Sfifilisof  35, 

3  eighths  of  32, 

12, 

24 

3  fourths  of  44, 

8  ninthspof  27, 

6. 

54 

4  fifths  of  30, 

6  sevenths  of  21, 

8, 

48 

Jrithmetica I  Froportioyu  125 

ARITHMETICAL  PROPORTION, 

Is  the  relation  between  two  numbers  with  respect  to  their 
difference. 

Four  quantities  are  in  arithmetical  proportion,  when  the 
difference  between  the^firstand  second  is  equal  to  the  diff'e- 
rence  between  the  third  and  fourth.  Thus,  4,  6,  T,  9,  are  in 
arithmetical  proportion,  because  the  difference  between  4 
and  6,  the  first  and  second,  is  2  ;  and  the  difference  between 
7  and  9,  the  third  and  fourth,  is  also  2. 


ARITHMETICAL  PROGRESSION, 

Is  a  continued  arithmetical  proportion,  or  it  is  a  series  oi 
numbers  which  increase  or  decrease  by  a  common  difference, 
as,  2,  4,  6,  8,   10,  •  2,  &c. ;  or,  20,   l6,  12,  8,  4,  &g. 

The  first  and  last  terms  are  called  extremes. 

In  any  series  of  numbers  in  arithmetical  progression,  the 
sum  of  the  two  extremes  is  equal  to  the  sum  of  any  two 
terms  equally  distant  from  them,   or  to   twice   the   middle 
term,  if  the  number  of  terms  is  unequal. 
Case  1. 

To  find  the  sum  of  all  the  terms,  whjsn  the  first  term,  the 
last  term,  and  the  number  of  terms,  are  given. 

liULE. 

Multiply  the  sum  of  the  two  extremes  by  half  the  number 
of  terms,  and  the  product  is  the  sum  of  all  the  terms. 

Example.  How   many   strokes  does  a  clock  strike  in  1^ 
;iiours  ? 

Operation, 
1  first  term. 

12  last  term. 

13  sum  of  the  two  ext^'^mes. 
6  half  the  number  of  terms* 

78  Ms. 

Case  2. 

To  find  the  number  of  terms,  when  the  first  and  last  terms, 
dnd  common  difference,  are  given. 

Divide  the  difference  of  the  extremes  by  the  common  dif- 
ference, add  I  to  the  quotient,  ana  it  wiii  be  the  number  of 
terms. 

L2 


12S  Arithmetical  Prognssion. 

Example.  If  a  man  gave  his  3roungest  child  20  dollars*, 
the  next  40,  and  so  on,  increasing  to  the  eldest^  who  had 
1 00,  how  many  children  had  he  ? 

Operation. 
100  last  term. 
20  first  term. 

com.  diff.  20)bO  difference  of  the  extremes* 

4+1=5,  Ms. 

Casp',  3. 
To  find  the  common  difference,   when  the  first  and  last 
terms,  and  number  of  terms,  are  given. 

Rule. 
Divide  the  difference  of  the  extremes  by  one  less  than  the 
number  of  terms,  and  the  quotient  will  be  the  common  diffe- 
rence. 

Example.  A  man  had  10  sons,  whose  ages  differed  alike  ; 
the  youngest  was  3  years  old,  and  the  eldest  48.  What  was 
the  common  difference  ? 

Operation. 
number  of  terms,  10  48  greater  extreme. 
1     3  less  extreme. 

divisor   9  )45  difference  of  the  extremes. 

5  Ms. 

Case  4. 

To  find  the  last  term,  when  the  first  term,  the  common 
difference,  and  number  of  terms,  are  given. 

Rule. 

Multiply  the  common  difference  by  that  number  which  is 
one  lens  than  the  number  of  terms  ;  then,  if  the  series  is  in- 
creahing,  add  the  first  term  to  that  product,  and  the  sum  will 
be  the  last  term  ;  but  if  the  series  is  decreasing,  subtract  that 
prv)duct  from  the  first  term,  and  the  remainder  will  be  the 
last  term. 

Example  1.  If  a  man  travels  4  miles  the  first  day,  and  7 
ihe  second,  and  so  on,  increasing  3  miles  each  clav.  ho^v 
far  will  he  travel  the  20th  dav  ? 


Geometrical  Frogresnon,  1 S7 

3  oommon  difference. 

19  one  less  than  the  number  of  terms. 

'  57  product. 

4  first  term. 

61  Ans* 
Example  2.  If  a  man  travels  6 1  miles  the  first  day,  and 
58  the  second,  and  so  on,  decreasing  3  miles  each  day  ;  how 
far  will  he  travel  the  20th  day  ? 

3  common  difference.  €1  first  term. 

19  one  less  than  the  number  of  terms«>       57  product* 

57  product.  4  Ans. 


GEOMETRICAL  PROGRESSION, 

Is  a  series  of  numbers  which  increase   or  decrease  by  a 

ommon  multiplier  or  divisor,  called  the  ratio,  as  2,  4,  8,  16, 

32,  64,  &c.  which  increase  by  the  common  multiplier  2 ;  or, 

486,  i62,  54,  18,  6,  2,  which  decrease  by  the  common  divisor 

3. 

In  any  series  of  numbers  in  geometrical  progression,  the 
product  of  the  two  extremes  is  equal  to  the  product  of  any 
two  terms  equally  distant  fiW  them,  or  to  the  square  of  the 
middle  number,  if  the  number  of  terms  is  unequal. 

Case  J. 
To  find  the  last,  or  aay  other  remote  term  ;  the  first  term, 
he  number  of  terms,  and  the  ratio,  being  given. 
Rule. 
Involve  the  ratio  to  that  power  which  is  one  less  tbanthe 
lumber  of  terms,   and   multiply  the  power  so  found  by  the 
iirst  term,  and  the  product  will  be  the  term  required. 

Example.  A  man  hired  a  laborer  for  one  year,  promising 
to  give  him  2  dollars  for  the  first  month,  4  for  the  second,  8 
for  the  third,  and  so  on ;  what  was  his  wages  for  the  last 
month  ? 

Operation,  The  ratio  is  2,  which  is  to  be  involved  to  its  11th 
I'ower,  because  12  is  the  number  of  the  term  sought.  The  11th 
power  ot*2  is  2048,  which  being  multiplied  by  2,  the  first  term,  gives 
1096  for  the  12th  term,  or  tUe  wages  of  the  last  month. 
Case  2. 
To  find  the  sum  of  the  series  ;  the  first  term,  the  last  term, 
and  the  ratio,  being  given. 


128  Mlgalion. 

Rule. 

Multiply  the  last  term  by  the  ratio,  from  the  product  sub- 
tract the  first  term,  and  divide  the  remainder  by  that  number 
which  is  one  less  than  the  ratio,  and  the  quotient  will  be  the 
sum  of  all  the  terms. 

Example.  According  to  the  terms  of  the  preceding  ques- 
tion, what  is  the  wages  of  the  laborer  for  the  whole  year  ? 

Operatioji.  The  first  term  is  2,  the  ratio  2,  the  number  of  terms 
12,  and  the  last  terra  4096,  as  found  by  Case  1.  Now,  I  multiply 
4096,  the  last  term,  by  2,  the  ratio,  and  the  product  is  8192  ;  tVom 
this  I  subtract  2,  the  first  term,  and  the  remainder  is  8190.  This 
is  to  be  divided  by  that  number  which  is  1  less  than  the  ratio  ;  but 
as  the  ratio  is  2,  the  number  which  is  1  less  than  it,  is  1  ;  and  8190 
divided  by  1,  gtves  8190  oollars  for  the  answer. 

Questions  on  the  foregoing. 


What  is  arithmetical  proportion  ? 

What  'S  arlthmetiQal  progression  ? 

What  are  the  extremes  ? 

i'o  what  is  the  sum  of  the  two  ex- 
tremes equal  '' 

How  do  you  find,  the  sum  of  all  the 
terras  ? 

JIow  do  you  find  the  number  of 
terms  ? 


How  do  you  find  the  common  diffe- 
rence ? 

How  do  you  find  the  last  term  ? 

What  is  geometrical  progression  ? 

To  what- is  the  product  of  the  ex- 
tremes equal  ? 

What  IS  the  rule  for  finding  the  last 
term  ? 

How  do  you  find  the  sum  of  the  se- 
ries ? 


ALLIGATION, 

Is  a  rule  which  teaches  how  to  mix  together  several  in- 
gredients of  differeat  values  or  qualities,  so  that  the  mixture 
maj  be  of  some  intermediate  value  or  quality. 
Case  1. 
To  fmd  the  value  or  quality  of  the  mixture,  when  the 
quantities,  and  values,  or  qualities,  of  the  several  ingredients 
ef  which  it  is  composed,  are  given. 

Rule. 
Multiply  the  quantity  of  each  ingredient  by  its  value  or 
quality  ;  then  say:  As  the  sum  of  the  quantities  of  the  seve- 
ral ingredients,  is  to  the  sum  of  the  several  products ;  so  is 
any  given  quantity  of  the  mixture,  to  its  value. 

Example.  A  mixture  being  made  of  5 lbs.  of  tea,  worth  f.v 
a  ih, ;  9lbs,  worth  bs,  6d,  a  lb,;  and  I4^l&s.  worth  5s.  lOt^.  a 
lb, :  what  is  a  lb.  of  it  worth  ? 

Operalion, 
lbs.         (L 
5    X   84===  420 
9    Xl02=  918 

14-5X  ro=ioi5 

t?um  of  the -™« 

quantities,  28-5  )2353  sum  of  the  product.«i> 


Alligation.  IS^ 

Ih.  d.        lb. 

!  Therefore,  28-5  :  2353  ::  \  i  82J(£.=65.  lOJ+rf.  dns. 

Explanation.  I  set  doivn  the  quantity  or  number  o^Bs.  5,  9,  and 
14§,  ui  a  convenient  coll  mn  for  addition,  expressing  the  halt*  lb,  by 
a  decimal,  for  greater  convenience.  Opposite  to  each  quantity,  I 
set  its  value,  reduced  to  pence,  because  the  value  of  one  sort  was 
partly  pence;  84,  102,  and  70.  I  then  set,  in  a  third  column,  the 
several  products  ibrmed  by  multiplying  each  quantity  by  its  value, 
420,  918,  and  1015.  1  then  find  the  sum  of  the  quantities,  which  is 
2S'5lbs.f  and  the  sum  of  the  products,  which  is  2353  ;  and  say,  a^ 
28*5,  the  sum  of  the  quantities,  is  to*?353,  the  sum  of  the  products ; 
so  is  1,  the  given  quantity,  to  its  value.  Which  proportion  being 
worked  out,  gives  82rf.  2q.,  or  65.  10(/.  2q,  for  the  value  of  a  lb. ; 
which  is  the  answer.  v 

Case  2. 

When  the  values  of  several  ingredients  are  given,  to  find 
how  much  of  each  will  make  a  mixture  of  a  given  value. 

Rule. 

1.  Set  the  values  of  the  ingredients  in  a  column  under  each 
other,  and  the  value  of  the  mixture  at  the  left, 

2.  Consider  which  of  the  ^;alues  of  the  ingredients  are 
greater  than  that  of  the  mixture,  and  which  less  ;  and  connect 
each  greater  with  one  less,  and  each  less  with  one  greater. 

3.  See  what  the  difference  is  between  the  value  of  eack 
ingredient  and  that  of  the  mixture,  and  set  down  that  difte- 
rence  opposite  to  the  value  with  which  such  ingredient  is 
connected. 

4.  Then,  if  only  one*difference  stands  against  any  value,  that 
will  be  the  quantity  belonging  to  that  value  ;  if  more  than 
one,  their  sum  will  be  the  quantity. 

JVote  I.  If  all  the  given  values  of  the  ingredients  are  greater  or  less  than 
that  of  the  mixture,  they  must  be  linked  with  a  cypher, 

JVo^e  2  Questions  of  this  kind  will  admit  of  as  many  answers  as  there  can 
be  different  modes  of  connecting  the  values,  or  of  dividing  them  by  a  common 
divisor,  or  multiplying  them  by  a  common  multiplier;  for  which  reason  they 
*re  called  indeterminate ^  or  unlimited  problems. 

Example.  How  much  oats,  a't  2s.  6^^.,  barley  at  3s.  %d** 
corn  at  4s.,  and  rye  at  4s.  %d.  per  bushel,  must  be  mixed  to- 
gether, that  the  compound  may  be  worth  3s.  lOrf.  per  bushel  ? 

Operation, 
hush. 

10  oats,  ^ 
2  barley,     f    ^^^^ 
2  corn,        f 

^  \  ^e,^      16.  rye,  J 


130  %Blligation, 

Explanation, 

1.  I  set  dowo  46r?.  the  value  of  the  mixture,  and  at  the  right  of  it, 
in  a  column,  3i)i.  the  vahie  of  the  oats,  44d.  that  oi*  the  barley,  48(i. 
-|hat  of  the  corn,  aivJ  56d.  that  of  the  rye. 

2.  I  con5>ider  v'.^hich  of  them  are  greater,  and  which  less  than  46, 
the  value  of  the  inixt\ire,  and  connect  them  accordingly,  each  great- 
er with  one  less,  and  each  less  with  one  greater ;  that  is,  48,  a 
greater,  with  44,  a  less,  and  56,  a  greater,  with  30,  a  less. 

3.  I  see  what  the  difference  is  betvveen  the  value  of  each  ingre- 
dient, and  that  of  the  mixture  ;  and  I  find  that  16  is  the  difference 
between  30  and  46.  So  I  >et  down  16  opposite  to  56,  with  which 
30  is  connected.  2  is  tht  difference  between  44  and  4^ ;  and  I  set 
down  opposite  to  48,  with  which  44  is  connected.  2  is  the  diffe- 
rence between  48  and  46  ;  and  I  set  down  2  opposite  to  44,  witli 
which  48  is  connected.  10  is  the  differenee  between  55  and  46  ; 
and  I  set  down  10  opposite  to  30,  with  which  56  is  connected. 

4.  Now,  as  I  have  only  one  difference  opposite  to  each  value,  that 
is  the  quantity  belonging  to  that  value  ;  that  is,  there  must  be  IG 
bushels  of  oats,  2  of  barley,  2  of  corn,  and  16  of  rye. 

Again:  The  operation  ijaay  be  varied,  and  a  different  answei 
produced,  by  connecting  the  values  in  a  different  manner,  as  follows: 


d.   d. 

hush. 

rso-^ 

2  oats, 

"^ 

44      ^ 

10  barley, 
16  corn. 

>  Ans, 

J  6^ 

2  rye, 

^ 

And  again,  as  follows  : 

d.      d. 

bush. 

f30-s:>v        10+S=12  oats,        "^ 

[^56-^      16      =16  rye,         J 
And  so  on  indefinitely. 
Proof.  Case  1  and  2  prove  each  other. 

Case  3. 
When  the  whole  mixture  is  to  consist  of  a  certain  quantity 

Rule. 
Find  the  quantity  of  each  ingredient,  by  Case  2  ;  and  then 
3ay,  as  the  sum  of  the  quantities  thus  found,  is  to  the  giver 
quantity;  so  is  the  quantity  of  each  ingredient  thus  found,  U 
the  quantity  required  of  eacli. 

Example.  How  much  oats,  at  2s.  6d,,  barley  at  3s.  Sd. 
corn  at  4s.,  and  rye  at  4s.  Sd.  per  bushel,  must  be  mixed  to- 
gether, to  form  a  mixture  of  90  bushels,  worth  Ss.  10c?.  pei 
bushel  ? 


Mligation<r 

Operation. 

bush. 

10  oats,  ^ 

barley,  T 

2  corn,  f 

rye,  J 


131 


■1 


Ans, 


this  example. 


*^nsr 


30  total.    Therefore* 
:     30,  oats, 

:       6,  barley  &  corn  each, 
:     48,  rye, 

proceed  as  before  ;  and  find  that  10 
bushels  of  oats,  2  of  barley,  2  ©f  corn,  and  16  of  rye,  would 
make  a  mixture  worth  3s.  \0d.  per  bushel.  But  ;  0+2+24- 1 6 
is  only  SO,  whereas  I  wanted  the  whole  to  be  90  bushels. 
So  I  say,  as  30,  the  sum  of  the  quantities  thus  found,  is  to 
90,  the  given ^  quantity ;  so  is  10,  the  quantity  of  oats  thus 
found,  to  30,  the  quantity  of  oats  required ;  and  so  of  the 
rest. 

Case  4. 
When  one  of  the  ingredients  of  which  the  mixture  is  com- 
posed, is  limited  to  a  certain  quantity. 

Rule. 
Find  the  difference  between  the  values  of  the  several  in- 
gredients, and  that  of  the  mixture,  and  arrange  them  as  in 
Case  2  ;  and  then  say,  as  the  difference  which  stands  oppo- 
site to  that  ingredient  whose  quantity  is  given,  is  to  the  rest 
of  the  differences  severally;  so  is  the  quantity  given,  to  the 
several  quantities  required. 

Example.  How  much  oats,  at  2s.  6d,  per  bushel,  barley  at 
3s.  8c/.,  and  corn  at  4s.,  must  be  mixed  with  24  bushels  of  rye, 
at  4s.  Sd.  that  the  mixture  may  be  worth  3s.  lOd.  per  bushel  ? 
Operatio7i. 
hush. 
10  oats, 
2  barley, 
2  corn, 
16  rye. 
Then,   16  :  fo  : :  24  :  15,  oats; 

3,  barley  &  com  each. 

In  this  example,  I  proceed  as  before  ;  and  find  that  10  bushels  of 
oats,  2  of  barley,  2  of  corn,  and  16  of  rye,  weuld  make  a  mixtiier 
worth  3s.  lOd.  per  bushel.    But  as  there  is  to  be  24  bushels  of  rye, 


d.       d. 

16  :   10  ::  24 
16  :     2  ::  24 


Ms. 


132  Position. 

instead  of  16,  the  quaotity  ol  each  of  the  other  simples  must  be  in- 
creased proportionally.  o  I  say,  as  16.  the  difference  which  stands 
©pposite  to  tht  rye,  is  to  10,  the  tiifference  which  stands  opposite  to 
the  oats  ;  so  Is  24,  the  gir^n  quandty  ot  the  rye,  to  15,  the  quantity 
of  oats  required  ;  ^md  so  .  f  tht  resu 

Questions  on  the  foregoing. 
What  is  animation  ?  ^!    Nfui    tlit-    values    are    connected. 


Whnt  »s  the  first  case  r  ,Vhat  is  the 
rme  ? 

What  is  the  secoix'    rs.-^ 

How  do  vou  aiT£K^-  i.;ie  severa'.  va- 
lues ? 

In  wiiat  manner  do  you  connect 
theai  ' 

What  is  to  be  doi  e,  if  all  the  values 
of  the  ingredievits  are  greater,  or 
all  less  than  that  of  the  mixture  * 


w'ha'i   's  io  be  done  ntxt' 
Wlveii  or.e  diiFeience  sta.  ds  opposite 

,o  .  iiy  ^'it'^e,  wiiMt  is  t:  «,  answer  ? 
Wlu  t    v.l.ef.  there  are  more  r 
WhH>.  •  r    oucstions  of  this  kind  call* 

^'d,  rji^  why  '. 
Of  'ow  r.-'any  arswei?   do  tl  ey  ad- 

mi>  ? 
What  is  the  ijiethod  of  proof? 
What  is  the  third  case  ?  The  rule  ? 
The  fourth  case  ?  i  he  rule  i 


POSITION, 

Is  a  tnethod  of  perfornfiing  certain  questions,  by  the  sup- 
position ot  false  numbers,  by  working  with  ^'i^iich/^the  true 
numbers  are  found. 

SINGLE  POSITION, 

Is  that  by  which  a  question  is  performed  by  means  of  one 
supposition  only. 

J\i"ote.  Questions  which  have  their  results  proportional  to  their  suppositions, 
belong  to  this  mle. 

P.ULE. 

Take  any  number,  and  perform  the  same  operation  with  it, 
as  is  described  in  the  question  ;  and  then  say,  as  the  result 
of  said  operation  is  to  the  number  taken,  so  is  the  result  in 
the  question,  to  the  number  sought. 

Example.  A  person,  after  spending  |  and  i  of  his  money, 
has  ^6o  left ;  what  had  he  at  first  ?   - 
Operation. 
Suppose  he  had  £  1 20.     Then, 

£.        £. 

J  of  120  is  40 

and  i  of  120  is  SO 

their  sum  is  ^TO,  which  beingtaken  from 
dei£0,  leaves  £'50.     Then,  50  :  120  ::  60  :  14^£.  dns. 
Proof, 
J  of  144  is  48 
i  of  144  is  36 

their  sum  is  ^684,  which  being  snbtract- 
ed  from  ^144,  leaves  ^60,  as  by  the  question. 


Position.  133 

DOUBLE  POSITION, 

Is  that  by  which  a  question  is  performed  by  means  of  two 
suppositions. 

JYote  Questions  whicli  have  their  results  not  propprtional  to  their  suppo- 
sitions, belong  to  this  rule. 

Rule. 

1.  Take  any  two  numbers,  and  proceed  with  each  of  them 
separately  according  to  the  conditions  of  the  question,  as  in 

■  single  position  ;  and  find  how  much  each  result  is  different 
from  the  result  mentioned  in  the  question,  calling  these  dif- 
ferences the  errors,  noticing  also  whether  the  results  are  too 
great  or  too  little. 

2.  Multiply  tlie  first  supposition  by  the  last  error,  and  the 
last  supposition  by  the  first  error. 

3.  If  the  errors  are  like,  divide  the  difference  of  the  pro- 
ducts by  the  difference  of  the  errors  ;  but  if  unlike,  divide 
the  sum  of  the  products  by  the  ^rn  of  the  errors  ;  and  the 
quotient  will  be  the  answer,  or  true  number  sought. 

A'^ote.  The  errors  are  said  to  be  like,  Avhen  they  are  both  too  ,J^r(?at,  or 
IboUi  too  little  ;  but  ujilike^  when  one  is  too  great,  and  the  other  too  little. 

Example  1.  What  nuir^ber  is  that,  which  being  multiplied 
by  6,  the  product  increased  by  18,  and  the  sum  divided  by  9, 
the  quotient  will  be  iiO  .^ 

Operation. 

1.  Suppose  it  to  be  18.  Then,  18x6  is  108,  and  18  added 
to  108  is  126,  and  126  divided  by  9  is  14.  But  instead  of 
14,  it  ought  to  be  20,  according  to  the  terms  of  the  question  ; 
therefore  the  error  is  6  too  little. 

Again  :  Suppose  the  number  to  be  30.  Then,  30x6  is 
180,  and  18  added  to  180  is  198,  and  198  divided  by  9  is  22. 
But  it  ought  to  be  20  ;  therefore  the  error  is  2  too  great. 

2.  Next,  r  multiply  13,  the  first  supposition,  by  2,  the  last 
error,  and  the  product  is  S6  ;  and  I  multiply  50,  the  last  sup- 
position, by  6,  the  first  error,  and  tlie  product  is  \  80. 

3.  To  know  whether  to  take  the  sum  or  dlilerence  of  these 
products  and  errors,  for  division,  I  consider  whether  the  er- 
rors are  like  or  unlike.  As  one  was  too  great  and  the  other 
too  little,  they  are  unlike  ;  and  1  take  the  sums.  The  sum  of 
36  and  180,  the  products,  is  216,  which  is  the  dividend  ;  and 
the  sum  of  6  and  2,  the  errors,  is  8,  which  is  the  divisor. 
And  21 6  divided  by  8,  gives  27  for  the  true  number  sought. 

Trnof,  2*^x6  is  l62,  and  18  added  to  162  is  180,  and  180 
divided  by  9  is  20,  according  to  the  terms  of  the  question. 

M 


134  Fermutation. 

Example  2.  A  man  left  10000  dollars  to  his  two  sons,  one 
aged  11,  and  the  other  l6,  to  be  divided  in  such  a  manner 
that  their  respective  shares  being  put  out  at  simple  interest 
at  4  per  cent  per  annum,  should  amount  to  equal  sums  when 
they  come  of  age.  "  What  are  the  shares  ? 

Operation. 

1.  Suppose  the  youngest  to  have  4000  dollars ;  then  the 
eldest  will  have  6000.  The  interest  of  4000  dollars,  at  4  per 
cent,  for  10  years,  is  1600  dollars  ;  which  makes  the  sum  of 
the  youngest  5600  dollars.  The  interest  of  6000  dollars  for 
5  years,  is  1200  dollars  ;  which  makes  the  sum  of  the  eldest 
7%00  dollars.  The  sum  of  the  youngest,  therefore,  is  1600 
dollars  too  little  ;  which  is  the  first  error. 

Again  :  Suppose  the  youngest  to  have  4500  dollars  ;  then 
the  eldest  will  have  5500;  and  the  amount  of  their  shares 
will  be  6300  and  6600,  which  makes  the  sum  of  the  youngest 
still  too  little  by  300  dollars. 

2.  Next,  the  suppositions  multiplied  by  the  errors,  are,. 
40(0x  00=1200000,  and  4500x  1600=7200  00  ;  and  the 
difference  of  the  products  is  6000000,  which  being  divided 
by  1300,  the  difference  of  the  errors,  gives  8461 5-3846,  for 
the  share  of  the  youngest ;  and  this  subtracted  from  Si 0000, 
gives  So384'6l54,  for  the  share  of  the  eldest. 

.  Froof,  To  prove  the  operation,  the  interest  of  S4615-3846 
for  10  years,  at  4  per  cent,  is  gl846*15o8,  which  added  ta 
the  principal,  makes  the  amount  S946 1-5384  ;  and  the  in- 
terest of  ]S5384'6 1 54  for  5  years  at  4  per  cent,  is  S  1076^9^30, 
which  added  to  its  principal,  makes  the  amount  S6461'5384  ; 
so  that  the  sums  are  equal,  according  to  the  terms  of  the 
question. 

Questions  on  the  foregoing. 


What  is  position  ? 
What  is  single  position  ? 
W  hat  questions  belong  to  single  po- 
sition ;' 
\\  ]\Rt  is  the  rule  ? 
W'hat  is  d-mble  pcsitton  ? 
What  questions  l)(,4ong  to  this  rule  ? 
What  is  the  first  thing  to  be  done  ? 


When  are  the  errors  said  to  be  like 

or  unlike  ? 
When   you   have  found  the  errors, 

wliRt  is  to  be  done  next  ? 
If  the  errors  are  like,   what   is  your 

divisor  ?  Your  dividend  ? 
It  they  are  unlike,  \Aiht  ^ 


PERMUTATION, 

Is  the  changing  of  the  position  or  order  of  things,   or   the 
showing  of  how  many  diSerent  ways  they  may  be  placed. 


Combination,  135 

Rule. 

.Multiply  all  the  terms  of  the  natural  series  together,  from 
1  up  to  the  givea  number,  and  the  last  product  will  be  the 
answer. 

Example.  How  many  days  can  7  persons  be  placed  in  a 
different  position  at  dinner  ? 

Operation,  1x2  is  2,  and  2x3  is  6,  and  6x4  is  24,  and 
24x5  is  120»  and  120x6  is  720,  and  720x7  is  5040  ;  which 
is  the  answer. 


m^  COMBINATIONT, 

IP     Is  the  showing  of  how  many  diiFerent  ways  a  less  number 
of  things  may  be  combined  out  of  a  greater. 

Rule. 

1.  Multiply  all  the  terms  of  the  natural  series  together, 
from  1  up  to  the  number  to  be  combined,  and  make  this  pro- 
duct the  divisor. 

2.  Take  another  series  of  numbers,  of  as  many  places,  be- 
ginning with  the  number  out  of  which  the  combination  is  to 
!)e  made,  and  decreasing  continually  by  1  ;  and  multiply 
them  t(»gether  for  a  dividend. 

3.  Divide  the  dividend  by  the  divisor,  and  the  quotient 
will  be  the  answer. 

Example.  How  many  combinations  can  be  made  of  6  let- 
ters out  of  10? 

Operation. 
1x2x3x4x5x6=720,  divisor; 
10x9x8x7x6x5==15l£00,  dividend. 
Then,  151200  divided  by  720,  gives  2  10,  for  the  answer. 

Questions  on  the  foregoing. 
What  is  permutation  ?  1 1  How  do  you  form  your  dmsor  ? 

What  is  the  rule  j"  |j  How,  your  dividend  \ 

What  is  combination  ?  What  is  the  answer  ? 

Exercise  77. 

Divide  2  iuto  3  such  parts,  that  the  sum  of  their  squares 
shall  be  1^,  Ans.  1,  §,  and  §. 

Divide  1  into  3  such  unequal  parts,  that  the  sum  of  their 
squares  shall  lack  |  of  being  i.  Jlns,  |,  J,  and  |. 

Divide  2  into  3  such  parts,  that  they  shall  have  the  ratio  of 

5,  2,  and  1.  Jins,  \\,  |,  and  ^. 
Divide  2  into  3  such  parts,  that  they  shall  have  the  ratio  of 

6,  3,  and  1.  Ms,  U,  },  and  j. 
Divide  2  into  3  such  parts,  that  the  sum  of  their  squares 

«hall  lack  ^  of  If.  Ms,  1,  f ,  and  §. 


V 


* 


136  Simple  Interest  by  JDeciniah, 

Divide  2  into  3  such  unequal  parts,  that  the  sum  of  their 
squares  shall  be  |  more  than  H.  Jlns,  1,  f,  and  ^. 

Divide  1  into  3  such  unequal  parts,  that  the  sum  of  their 
squares  shall  lack  J^  of  i,  Jlns,  f ,  a,  and  i. 

Divide  1  into  2  such  part«,  that  the  difference  of  their 
squares  shall  be  §.  Ans,  |,  and  §. 

Divide  I  into  2  such  parts,  that  the  difference  of  their 
squares  shall  be  \,  Jlns.  |,  and  ^. 

Divide  5  into  2  parts,  in  the  ratio  of  1,  and  2. 

Jlns,  If,  and  3 J. 

There  are  2  numbers,  and  the  difference  between  their 
sum,  and  the  sum  of  their  squares,  lacks  y^  of  being  i ;  what 
are  the  numbers  ?  Jins,  i,  and  %, 

Divide  a  shilling  into  2  parts,  so  that  one  part  shall  be  one 
farthing  more  than  the  other.  Jlns,  ^\d,  and  5|6?. 

Divide  3s.  Ad,  into  2  parts,  so  that  one  part  shall  be  2\d, 
more  than  the  other.  Ans.  ]s,  9^d  and  Is.  6fc?. 

Divide  a  dollar  into  three  parts,  so  that  the  largest  shall 
be  8  cents  more,  and  the  smallest  7  cents  less,  than  the  mid- 
dle part.  Jins.  41  cents,  33  cents,  and  26  cents. 


SIMPLE  INTEREST,  BY  DECIMALS. 

Mdtio  is  the  simple  interest  of  ^1.  or  i  dollar  for  1  year, 
at  any  given  rate,  expressed  as  the  decimal  of  a  ^.  or  a  dol- 
lar. Thus,  5  per  cent,  is  -05  ;  six  per  cent,  '06  ;  six  and  a 
half  per  cent,  '065,  &c. 

Case  1. 

The  principal,  time,  and  ratio  given,  to  find  the  interest. 
Rule. 

Multiply  the  principal,  time,  and  ratio,  continually  toge- 
ther, and  the  product  will  be  the  interest. 

Example.  What  is  the  interest  of  365^.  5s.  for  10  years 
and  6  months,  at  6  per  cent  ? 

Operation,  The  principal  is  £365*:25,  the  time  is  10'5z/r. 
and  the  ratio  is  '06,  which  being  multiplied  continually  to- 
gether, tiie  answer  is  ^230-1075,  or  ^230  ..  2  ..  1|. 

J\  ate.  To  find  the  amount,  add  the  interest  to  tlie  principal. 
Case  2. 

The  amount,  time,  and  ratio  given,  to  find  the  principal. 
Rule. 

Multiply  the  time  by  the  ratio,  and  add  1  to  the  product 
for  a  divisor,  by  which  divide  the  amount,  and  the  quotient 
will  be  the  principal. 


Simple  Interest  by  Decimals.  137 

Example.  What  principal  will  amount  to  SSIOL,  in  6 
years,  at  4^  per  cent  ? 

Operation*  The  time  is  6,  and  the  ratio  is  '045  ;  and  their 
product  is  -27  ;  to  which  I  being  added,  the  divisor  is  1'27. 
And  the  amount,  ^38 1(,  being  divided  by  1-27,  the  quotient 
is  ^3000,  which  is  the  answer. 

JYote.    This  case  is  the  same  as  discount.     The  principal  found,  being  the 
same  as  the  present  worth. 

Case  3. 
The  amount,  principal,  and  time  given,  to^  find  the  ratio. 

Rule. 
Subtract  the  principal  from  the  amount,  the  remainder  will 
be  the  interests     Divide  the  interest  by  the  product  of  the 
time  and  principal,  and  the  quotient  will  be  the  ratio. 

Example.  At  what  rate  per  cent  will  543  J.  amount  to 
705^.  18s.  in  5  years  ? 

Operation, 
Principal,  543         amount,  705*9 
time,       5     principal,     543* 

2715)  Int.  l62-90(-06  ratio  of  6perceiit 

162-90  Ms. 

Case  4. 
The  amount,  principal,  and  rate  per  cent  given,  to  find 
the  time. 

Rule. 
Find  the  interest,  and  divide  it  by  the  product  of  the  prin- 
cipal and  ratio  ;  the  quotient  will  be  the  time. 

Example.  In  what  time  will  543d£.  amount  to  705^.  18^, 
at  ()  per  cent  ? 

Operation. 
Principal,  543         amount^  705*9 
ratio,  '06       principal,  543* 

32-58)  Int.  162-90(5  years.  Ms. 

162-90 


COMPOUND  INTEREST  BY  DECIMALS. 

Ratio  is  the  amount  of  ^l,  or  SI,  for  1  year,  expressed  m 
a  decimal  form. 

Case  1. 
The  principal,  rate,  and  time  given,  to  find  the  amount. 

Rule. 
1.  Involve  the  ratio  to  such  a  power  as  is  the  same  witk 
flie  number  of  years. 

M2 


138  Vomponml  Interest  hj  IJccimah. 

2.  Multiply  the  power  so  found,  by  the  principal,  ar^l  the 
product  will  be  the  amount. 

J\oie.  Having  t'oui^d  the  amount,  subtract  the  pi  mcipul  from  it,  v.  ^  Iht 
remainder  will  be  the  compound  inteiesi. 

Example.  What  Is  the  amount  of  ^3^-1  for  4  yeai  -  it  5 
per  cent,  compound  interest  ? 

Operation.  The  ratio  is  1*05  ;  of  whitL  the  -th  power  (be- 
cause 4  is  the  nuniber  of  years)  is  l'2l550^>  5  ;  and  this  being 
multiplied  by  300,  the  principal,  gives  ^^36 4* 65 1875,  for  the 
amount,  which  is  the  answer. 

^Case2. 

The  aiiiount,  rate,  and  tiiue  given,  to  find  the  principal. 

EULE. 

Divide  the  aBiouat  given  by  the  ratio,  involved  to  such 
power  as  is  the  same  as  the  given  number  of  years,  and  the 
quotient  will  be  the  principal. 

Example.  What  principal,  at  5  per  cent,  compound  inte* 
rest,  for  4  years,  will  amount  to^364*65  1875  ? 

Operation.  The  ratio  is  L\'05,  and  its  4th  power  Ll- 
•21550625,  and  :^64-651875  divided  by  i-£l 550625,  gives 
SOoA.  for  the  answer. 

r'  J^te   Tliis  case  is  the  same  as  discount  at  compeund  iriterest,  the  principal 
found,  being  the  same  as  the  present  wortli* 

Cas:3. 
The  princ  pal,  rate,  and  aitiount  given,  to  find  the  time. 

Divide  the  amount  by  the  principal,  then  involve  the  ratio 
till  it  equals  the  quotient,  and  the  number  (if  involutions  will 
be  the  same  as  the  number  of  years. 

Example.  In  what  time  will  jL4oO  amount  to  X520  93125^, 
at  5  per  cent,  compound  interest  ? 

Operation.  The  amount,  L520«93125,  being  divided  by 
X450,  the  principal,  gives  M  67625,  for  the  quotient.  The 
ratio,  1*05  involved  to  the  3d  power,  is  1-157625,  which, 
equals  the  quotient*     So  the  arlsw^er  is  3  years. 

Case  4. 
The  principal,  amount,  and  time  given,  to  find  the  rate  per 
cent. 

Rule.  '' 

Divide  the  amount  by  the  principal,  and  extract  such  root 
of  the  quotient  as  is  denoted  by  the  number  of  years;  which 
"oot  will  be  the  ratio. 

Example.  At  what  rate  percent  will  45(DX.  amount  to 
/.o'20'93i25j  in  three  years  ? 


Annuities  at  Simple  Interest  1 39 

Operation.  5':>0-93I26    divided  by  4.'.0,  gives   M57625; 
vA  the  cube  root  ot  M576i5,  is  l'05,  the  ratio  of.')  percent, 

QlESTIONS  ON  THE  FOKEGOING. 

In   couipOMud   interest  by  decimals, 

what  is  meai.i  by  the  7'atio  ? 
How  d»}ou  find   the  amount?  The 

compouiid  interest  ? 
i  he  principal  ?  the  time  ?  the  rate 

per  cis  it  ? 
Which  case  is  the  same  as  discount, 

and  why  ? 


In  sim])le  interest  by  decimals,  what 

is  meant  by  ratio '^ 
How  do  \ou  find  the  i.iteiest  ? 
How,  the  pricipal  \  flov,  the  ratio  ? 

The  time  ? 
Which  case  is  the  same  as  discount, 

and  why  ? 


ANNUITIES. 

An  annuity  is  a  sum  of  nionej  payable  every  year,  for  a 
number  of  years,  or  forever* 

When  the  annuity  is  not  paid  as  it  becomes  due,  it  is  said 
to  be  in  arrears. 

AVhen  the  annuity  is  not  to  begin  till  after  a  certain  time 
has  elap  ed,  it  is  said  to  be  in  reversion. 

The  sum  of  all  the  annuitiers  in  arrears,  together  with  the 
interest  due  upon  each,  is  called  the  amount. 

If  an  annuity  is  to  be  bought  otf,  or  paic^  all  at  once,  the 
price  which  ought  to  be  paid  for  it,  is  called  the  present  worth. 
ANNUITIES  AT  SIMPLE  INTEREST. 

Case  1. 

To  find  the  amount  of  an  annuity  at  simple  interest. 
Rule. 

1.  Make  the  first  term,  and  1  the  common  difference,  of 
a  series  of  numbers  in  arithmetical  progression,  and  make  the 
number  of  terms  one  less  than  the  number  of  years  ;  and  find 
the  sum  of  the  series. 

3.  Multiply  that  sum  by  one  year's  interest  of  the  annuity, 
and  the  product  will  be  the  whole  interest* 

3.  Multiply  the  annuity  by  the  number  of  years,  and  add 
the  whole  interest  so  found,  and  the  sum  will  be  the  amount 
sought. 

J\%te.  The  reason  for  making  the  number  of  terms  in  the  arithmetical  ae- 
rie s  o),e  less  ihai^  the  number  ot  years,  is,  that  there  is  no  interest  due  upon 
the  last )  ear's  an;  uity 

Example.  What  is  the  amount ^of  an  annuity  of  700  dol- 
lars for  6  years,  allowing  simple  interest  at  7  per  cent  ? 
Op,  ration. 
1+24-3+4+5=15  sum  of  the  series. 

49  i  te.  e^t  of  7(''0  dollars  for  I  year. 
7^->5  whole  interest. 
6x7QO=»=42eo  six  annuities. 
S4935  amount. 


148  Annuities. 

J\''ote.  The  reason  of  this  operation  will  appear,  if  we  eonsider  that  at  the 
end  of  G  years,  there  is  due  tlie  first  year's  annuity,  700  dollars,  and  its  in- 
terest for  5  years,  that  is  5  times  49  doilars,  wh)ch  is  '245  dollars  ;  the  second 
year's  annuity,  700  dollars,  and  its  interest  for  4  years,  196  dollars  j  ttie  third 
year's  annuity,  700  dollars,  and  its  interest  for  3  years,  147  dollars;  the 
fourth  year's  annuity,  700  dollars,  and  its  interest  for  2}earf,  98  dollars;  the 
fifth  year's  annuitj ,  700  dollai-s,  and  its  interest  for  1  year,  49  dollars;  and 
the  sixth  year's  annuity,  700  dollars  :  all  which  added  together,  makes  4935 
dollars,  as  before. 

Case  2. 
To  find  the  pres«it  worth  of  an  annuity  at  simple  interest. 

Rule. 
Find,  as  in  discount,  the  present  worth  of  ea«h  payment  by 
itself,  allowing  discount  to  the  time  it  becomes  due,  and  the 
sum  of  all  these  will  be  the  present  worth  sought. 

Example.  What  is  the  present  worth  of  an  annuity  of  100 
dollars,  to  continue  5  years,  at  6  per  cent  per  annum,  simple 
interest  r 

Operation.  [annuity. 

94'SS96,  present  worth  of  the  1st  year's 
89'  857,  do.  2d     do. 

84-74  5  r,  do.  3d     do. 

8-6451,  do.  4th  do. 

76-9230,  do.  5th  do. 


106  : 

100  : 

:  100 

ll^Z   : 

100  : 

:  100 

118  : 

100  : 

:  100 

124  : 

100  : 

:  100 

ISO  : 

100  : 

:  100 

Ans,  S425-9391,  present  worth  of  the  whole. 

ANNUITIES  AT  COMPOUND  INTEREST. 

Case  1. 

To  find  the  amount  of  an  annuity  at  compound  interest. 

Rule. 

1.  Make  1  the  first  term  of  a  series  of  numbers  in  geome- 
trical progression,  and  the  amount  of  L\  or  gl  for  1  jear,  at 
the  given  rate  per  cent,  the  ratio,  expressing  it  in  decimals. 

2.  Extend  the  series  to  as  many  terms  as  the  number  of 
years,  and  find  its  sum. 

3.  Muldply  the  sum  thus  found  by  the  given  annuity,  and 
the  product  will  be  the  amount  required. 

Example.  What  is  the  amount  of  an  annuity  of  200  dol- 
lars, for  5  years,  allowing  compound  interest,  at      per  cent  ? 

Operation,  The  first  term  is  I;  the  ratio  is  1-05.  The 
first  term  being  multiplied  by  the  ratio,  gives  f05forthe 
second  term;  and  that  being  multiplied  by  the  ratio,  gi^es 
1-025  for  the  third  term  ;  and  that  being  multiplied  by  the 
ratio,  gives  )-15762;)  for  the  fourth  term  ;  and  that  being 
multiplied  by  the  ratio,  gives  1-21550625  for  the  fifth  term. 


•Annuities.  141 

The  sum  of  these  five  terms  is  5-52563125,  which  being  mul- 
v  tiplied  by  200,  the  annuity,  gives  81105' 12625  for  the  answer. 

J\''ote.  '1  o  find  the  amount  for  additional  parts  of  a  year.    Having  found 
j     the  amount  for  the  whole  years,  find  the  interest  of  that  amount  for  the  givea 
parts  of  a  year,  and  add  it. 

Case  2. 
To  find  the  present  worth  of  an  annuity  at  compound  in- 
1    terest. 
I  Rule. 

1.  Take  the  amount  of  Xl  or  gl  for  1  year,  at  the  given 
rate  per  cent,  and  involve  it  to  that  power  which  is  the  same 
as  the  number  of  years,  for  a  divisor. 

2.  Divide  the  annuity  by  this  divisor,  subtract  the  quo- 
I  tient  so  found  from  the  annuity,  and  set  down  the  remainder 
X    for  a  second  dividend.  » 

S.  From  the  amount  of  L\  or  Si  for  1  year,  subtrac  t  1 
and  take  the  remainder  for  a  second  divisor. 

4.  Divide  the  second  dividend  by  the  second  divisor,  and 
the  quotient  will  be  the  present  worth  required. 

Example.  What  is  the  present  worth  of  an  annuity  of  40 
dollars,  to  continue  5  years,  discount  at  5  per  cent  per  an- 
num, compound  interest  ? 

Operation,  1^  The  amount  of  1  dollar  for  i  year  at  5  per 
cent,  is  1*0  5.  This  being  involved  to  the  5th  power,  because 
5  is  the  number  of  years,  is  1-3762815625,  which  is  the  first 
divisor. 

2.  The  annuity,  40  dollars,  being  divided  by  1-2762815625, 
gives  3 1-S4 104,  for  the  first  quotient ;  which  being  subtracted 
from  40,  the  annuity,  leaves  8-65896,  f&^r  the  second  dividend. 

3.  From  1-05,  the  amount  of  1  dollar  for  1  year,  I  subtract 
1,  and  the  remainder  is  -05,  which  is  the  second  divisor. 

4.  The  second  dividend,  8-65896,  being  divided  by  -05, 
the  second  divisor,  gives  S 173*  175 2  for  the  present  worth, 
which  is  the  answer. 

JSfote.  To  find  ihe  present  worth  for  additional  par<s  of  a  year:  Havinj^ 
found  the  present  worth  for  ihn  whole  years,  find  tl:fc  presti.t  wrrthof  that 
present  worth,  discount  being  allowed  fortlie  givtn  paits  of  a  yeaj*. 


PERPETUITIES, 

Are  annuities  which  are  to  continue  forever. 
Case  1. 

To  find  the  present  worth  of  a  perpetuity  at  compound  in- 


terest 


]  42  Ferpetuitieis. 

Rule. 
As  the  rate  per  cent,  is  to  1 00  ;  so  is  the  yearly  payment, 
to  the  present  worth. 

Example.  What  must  I  give  for  an  annuity  of  40  dollars, 
to  continue  forever,  discounting  at  5  per  cent,  compound  in- 
terest. 

Operation. 
5      :      100      :   :      40      :  JVote.  To  find  what  perpe- 

1  QQ  tuity  can  be  purchased  for  a 

given  sunij    say,  as   100  is  to 

the   rate   per  cent,    so  is  the 

5)4000  given  sura,  to  the  perpetuity  it 

will  purchase. 

SBOO  Ms. 

Case  2. 

To  find  the  present  worth  of  a  perpetuity  in  reversion. 

Rule. 

1.  Take  the  amount  of  Li  or  Si  for  1  year,  at  the  given 
rate  per  cent,  and  involve  it  to  that  power  which  is  the  same 
as  the  number  of  years  before  the  annuity  commences. 

2.  Multiply  the  power  so  found  by  the  given  interest  of 
Xl  or  SI  tor  1  year. 

3.  Divide  the  given  annuity  by  tlie  product  so  found,  and 
the  quotient  will  be  the  present  worth  required. 

Example.  What  must  I  giveLfor  a  perpetuity  of  §40  per 
annum,  to  commence  5  years  hence,  discounting  at  5  per 
cent  ? 

Operation,  The  amount  of  gl  for  1  year,  is  1*05  ;  which 
being  involved  to  the  5th  power,  is  i*276g8l5625 ;  and 
1-2762815625  multiplied  by  -05,  the  interest  of  Si  for  i 
year,  is  '©eSS  14078125,  which  is  the  divisor  ;  and  g40  being 
divided  by  it,  gives  8626*8209+,  for  the  present  worth,  or 
answer. 

Question*  A  man  has  left  an  estate,  wliich  will  yield  SI 00 
a  year  forever,  to  his  two  sons,  A  and  B.  A  is  to  enter  upon 
it  immediately,  and  have  the  use  of  it  1 5  years  ;  after  which 
B  is  to  have  it  forever.  Whose  portion  is  the  most  valuable, 
and  how  much  the  most,  discounting  at  5  per  cent,  compound 
interest  ? 

Operation.  The  value  of  A's  portion  is  equal  to  the  pre- 
sent worth  of  an  annuity  of  S 10')  for  1 5  years  ;  and  the  value 
of  B's  portion  is  equal  to  the  present  worth  of  a  perpetuity  ia 
reversion  of  SlOO,  to  commence  after  1 5  years. 

To  find  A's  amount,  I  proceed  as  follows : 


Perpetuities. 


143 


The  amount  of  gl  for  1  year,  is  1'05  ;  which  being  involv- 
ed to  the  15th  povi^er,  is  2*078928179+.     Next,  the  annuity, 
g'OO,  is  to  be  divided  by  this  power ;  which  being  done,  the 
[   quotient  is  48*  101 7098-1- .     This  quotient  being  subtracted 
I  from  100,  the  annuity,  gives  5 1 '898290 1 -|-   for  the   second 
dividend.     Next,  from  1'05,  the  amount  of  g    for  1  year,  I 
^    subtract  !• ;  and  the  remainder  is  -05,  for  the  second  divisor. 
\   Lastly,  1  divide  51-8982901+,  the  second  dividend,  by  -05, 
^  the  second  divisor,  and  the  quotient  is  gi 03 7*9658+,  the 
present  worth  of  A's  portion. 

To  find  B's  portion,  I  proceed  as  follows  : 
Here,  again,  I  take  i*05,  and  involve  it  to  the  l5th  power, 
which  is  2-07^9'i8l79+,  as  before.  Next,  1  multiply  this  by 
'05,  the  given  interest  of  S ]  for  1  y^ar,  and  the  product  is 
•10394640895+ ;  by  which  I  divide  D 100,  the  given  annuity, 
and  the  quotient  is  D962'0341+,  the  present  worth  of  B's 
portion.  A's  portion,  therefore,  is  D  1037*9658+  ;  and  B's, 
D96>>0341+.  Consequently,  A's  is  the  most  valuable  by 
75  dollars,  93  cents,  1  mill,  and  7  tenths  of  a  mill. 

Questions  on 

What  is  an  annuity  ? 

When  is  it  said  to  be  in  arrears  f 

When,  in  reversion  ? 

What  is  the  amount  ? 

What  is  the  present  worth  ? 

What  is  the  first  case  of  annuities  at 

simple  interest  ? 
In  working  that  case,  what  is  the  first 

thing  to  be  done  ? 
The  second  ?     The  third  ? 
*i     Of  how  many  terms  do  you  make 
1        your  arithmetical  series  ? 
I    Why  so? 

What  is  the  second  casr:  ?  the  rule  ? 
What  is  the  first  case  of  annuities  at 

compound  interest  ? 
In  working  that  case,  what  is  the  first 


THE  FOREGOING. 

thing  to  he  done  ? 
The  second?     The  third  ? 

How  do  you  find  the  amount  for  ad- 
ditional parts  of  a  year  ? 

What  is  the  second  case  ? 

In  working  the  second  case,  what  is 
the  first  thing  to  be  done  ?  The 
second  ?  the  third  ?  the  fourth  ? 

How  do  you  find  the  present  worth 
of  an  annuity  in  rcA  ersion  ^ 

How,  for  additional  parts  of  a  year  ? 

What  are  perpetuities  ? 

^^  hat  is  the  first  case  ?  The  rule  ? 

What  is  the  second  case  ? 

In  working  the  second  case,  what  is 
the  first  thing  to  be  done  ?  The 
second  ?  the  thiid  ? 


MENSURATION. 

A  superficies,  or  surface,  is  that  which  has  length  and 
breadth,  but  not  thickness.  It  is  called  a,  plane  supnficies, 
when  the  surface  is  even,  without  any  curvature ;  that  is, 
when  it  is  such,  that  if  you  take  any  two  points  in  the  surface, 
and  draw  a  straiglit  line  from  one  point  to  the  other,  the 
whole  of  that  straight  line  will  be  in  the  said  surface. 

The  area  is  the  whole  surface  enclosed. 


144 


Mensuration. 


A'^ote.  In  measuring  the  length  of  a  road,  the  chain  or  measuring  rod  must 
fee  kept  parallel  with  ihe  surface  of  the  ground,  however  irregular;  because 
the  traveller  cannot  move  horizontally,  but  must  go  up  and  down  all  the  hills. 
But  in  measuring  land,  the  measuring  rod  must  be  kept  parallel  to  the  hori- 
zon, or  upon  an  exact  level,  the  reason  for  which  is.  that  all  the  calcula- 
tions of  the  quantity  of  land,  are  calculations  of  the  areas  of  plane  figures. 
And  they  ought  in  justice  to  be  so ;  because,  although  a  piece  of  ground 
which  has  a  hill  in  it,  has,  in  reality,  more  surface  than  if  the  hill  was  re- 
moved, and  it  was  reduced  to  an  exact  level  ,  yet  nothing  more  can  grow 
upon  it;  for  the  stalks  of  grain  always  shoot  up  in  a  direction  perpendicular 
to  the  horizon,  and  not  perpendicular  to  the  surface  of  the  soil. 

A  plane  triangle  is  a  figure  bounded  by  three  straight  lines. 
Every  piece  of  ground,  bounded  by  right  lines,  may  be  di- 
vided into  triangles. 


B 


Example. 

Let  ABCDEF,  be  an  irregular  field 
of  six  sides.  Draw  a  line  from  A  to 
C,  from  C  1o  E,  and  from  E  to  A,  and 
it  will  be  uivided  into  four  triangles. 


Problem  1.  To  find  the  area  of  a  triangle* 
Rule  1.  Measure  one  side  of  the  triangle,  and  also  mea- 
sure the  perpendicular  distance  from  that  side  to  the  opposite 
angle. 

£.  Multiply  these  together,  and  half  the  product  will  be 
the  same. 

JVote  I  he  most  convenient  instrument  for  measuring  land,  is  the  chain. 
As  160  ^uaie  poles  make  an  acre,  the  chain  is  made  4  poles,  or  66  feet  in' 
length  ;  and  so,  10  square  chains  make  a>i  acre  i  he  chain  again  is  divided 
into  100  links  And  the  account  being  kept  in  chains  and  decimals  of  a 
chain,  is  reduced  to  acres  and  decimals  of  an  acre,  by  removing  the  decimal 
point  one  figure  to  the  left. 

Example. 


Let  ABC  be  a  triangular  field.  Let  the 
side  AB  be  ^5  chains  and  75  links,  and 
the  perpendicular  CD  1 6  chains  and  25 
links.  How  many  acres  does  the  field 
contain  ? 


Operation.  25-75x  16-25=418,4575,  and  half  of  that,  or 
209-21875  is  the  area  in  chains  ;  which  gives  20-921875  for 
the  area  in  acres,  M^iich  is  the  answer. 

Where  a  chain  cannot  be  had,  a  pole  may  be  used,  16j 
feet  long,  which  is  the  length  of  a  pole,  rod,  or  perch.    The 


Mensuration,  145 

fbove  field  being  measured  by  such  a  p©le,  the  side  AB 
Would  be  103  poles,  and  he  perpendicular  CD  6s  poles, 
i  These  multiplied  together,  make  6695  ;  and  the  half  of  that 
i  is  3347 1,  which  is  the  area  in  poles:  which,  divided  by  160, 
/  the  number  of  square  poles  in  an  acre,  gives,  as  before, 
)   20*921775  acres,  or  20  acres,  3  roods,  and  27^  poles. 

.'  J\/ote.  I  o  finO  the  point  i  1  the  side  \B,  wheiv  the  perpendicular  from 
I  the  a'tgle  C  will  taU  :  Maki-  a  cross  of  two  pu;c  s  f  wood,  which  shall  cross 
\  cacli  otlierat  eight  angles,  (thus  f ;)  whicli  may  eSsily  be  done  by  the  help 
of  H  coinmoii  carpv-nter's  square  Lay  the  cross  ujion  the  line  AB.  so  that 
ot«eof  the  piece>  -hall  coincide  with  ihat  line;  and  then  move  it  along  that 
line,  tiil  the  oihe-  piece  shall  point  to 'he  angle  C  ;  and  the  point  where  the 
cross  stands,  will  be  the  point  where  the  |te!j)C'  dicnlar  falls.  1(  is  not  sup- 
posed iliat  lh;s  method  wid  give  the  point  exaciiy,  hiu  near  enough  for  coni- 
iiion  ])in'poses. 

Where  a  perpendicular  cannot  be  conveniently  measured, 
work  by 

Rule  2. 
1.  Measure  all  the  sides  of  the  triangle,  add  them  together, 
and  take  half  the  sum. 

!2.  Subtract  the  sides,   one    by  one,   from  the  half  sum  S9 
found,  and  note  the  three  remainders. 

3.  Multipl;y  the  half  sum  and  the  three  remainders  all  to- 
- 'gather,   and  the  square  root  of  the  last  product   will  be  die 
area. 

Example. 
C 

A  Let  ABC  be  a  triangle,  of  whicli  the  side  AB 

B  is  103  pole^,  BC  77  poles,  and  CA  9.  poles  ; 

what  is  the  area  ? 
Operation.  1034-77+90  is  270,  half  of  which  is  135%  Next, 
103  from  .  35  leaves  32  ;  90  from  15  leaves  45,  and  77  from 
h 35  leaves  58.      I  hen,    135x32x45x58  is    11^75^200;  the 
square  root  of  which  is  3357'8,  for  the  area  in  poles,  or  20 
acres,  3  roods,  and  ST*8  poles. 

Problem  2.   To  find  the  area  of  a  field  which  has  four 
sides  parallel  to  each  other. 

Rule. 
Multiply  the  length  by  the  breadth,  and  the  product  will 
be  the  area. 

Example. 
ABE  F  Let  ABCD,  or  EFGH, 

be  a  field  of  four  parallel 
sides,  to  find  the  area. 
13  I  H  If  the  angles  are  right 

^ngle^,  as  in  the  figure  ABCD,  AB  or  CD  will  be  the  length, 
N 


s/ru 


1 46  Mensuration* 

and  AC  or  BD  the  breadth.     But  if  the  angles  are  not  rio;ht 
angles,  as  in  the  figure  EFGH,  measure  a  perpendicular  from 
one  side  to  the  other,  as  EI,  and  this  will  be  the  breadth. 
Let  AB  be  S..  chains,  and  AC  14  chains  ;  what  is  the  area? 
Operation-  32x14=4^8  chains,  or  44*    acres.  Jins, 
Again,  i  et  EF  be  32  chains,  and  El  14  chains;  the  an- 
swer is  the  same. 

A^'ote  If  the  sides  are  ?i#t  parallel,  divide  the  field  into  triangles,  as  be- 
fore; for  if  J  oil  multiply  the  length  by  the  breadth,  you  will  not  have  the 
true  area. 

Problem  3.  To  find  the  art' a  of  a  circle, 

i-ULE. 

Square  the  diameter,  and  multiply  that  square  by  the  de- 
cimal '7854,  and  the  product  will  be  the  area. 

Example.  If  a  rope,  3  rods  long,  be  tied  one  end  to  a 
horse 'ji  head,  and  the  other  end  lo  a  stake,  how  great  an 
area  of  grass  can  he  eat? 

Operation,  As  3  rods  is  ihe  radius,  or  distance  from  the 
centre  to  the  circumft'ren<:e,  6  rods  is  tht  dian»eter  of  the 
circle.  This  squared  s  36  ;  and  that  n  ulHplied  by  v854, 
is  2b'Jsi744  rods  or  polt  s,  the  area  of  the  Ciicle. 

Problem  4.  To  faid  the  diameter,  or  the  circumference,  one 
from  the  (Aher, 

RuLp.  As  lis  is  to  355,  so  is  the  diameter  to  the  circum- 
ference. 

Example.  What  is  the  circumference  of  a  circle,  whose 
diameter  is  315  rods  r 

Operation.  113  :  355  : :  315  :  98;>6-f  Jns. 

pROBi.EM  5.  lb  fmd  the  relative  proportion  of  similar 
figures. 

Rule.  The  areas  of  similar  figures  are  to  each  other  a& 
the  ;  quares  of  their  similar  dimensions. 

Example.  If  a  rope  3  rods  long  allow  a  horse  to  graze 
28*  744  rods  of  ground,  how  long  must  a  rope  be  to  allow 
him  to  graze  an  acre  r 

Operation,  28-t744  :  9  ::  160  :  50-9  94.  That  is,  as 
the  first  area  is  to  the  square  of  the  length  of  the  rope,  or  ra- 
dius of  the  circle,  so  is  the  second  area,  to  the  square  of  its 
radius;  which  being  found  to  be  50*9  9  ,  its  square  root  is 
7* !  3-f  rods,  which  is  the  length  of  the  rope  required. 

Problem  6.  To  measure  the ;  ei^ht  oj  a  tree,  or  other  object. 

Rule.  Set  up  a  pole  perpendicularly,  the  lengTh  of  which 
above  the  ground  is  krtown.  Go  to  the  foot  of  the  tree,  and 
aaake  amark  in  it  at  the  height  of  your  eye  from  the  ground. 


Mensuration. 


147 


and  make  a  mark  in  <he  pole  at  tjie  same  height.  Then  go 
backwaid  till  you  find  such  a  station  that  your  eye  shall  be 
exactly  in  a  range  with  the  top  of  the  |>ole  and  the  top  of  the 
tree,  and  also  in  a  range  with  the  marks  in  the  pole  and  in 
the  tr^ee.  Measure  the  d  stance  from  that  station  to  the  toot 
of  the  pole,  and  also  to  the  foot  of  the  tree.  And  then  say, 
as  the  distance  from  your  station  to  the  foot  of  the  pole,  is  to 
the  he  ght  of  the  pole  above  the  mark ;  so  is  the  distance 
from  your  station  to  the  foot  of  the  tree,  to  the  height  of  tie 
tree  above  the  mark.  Then,  add  to  the  height  so  found,  the 
distance  from  the  mark  to  the  ground,  and  the  sum  will  be 
the  true  height  of  the  tree. 

Example. 
^Kk  B         Let  AB  be  a  tree,  the  height  of  which 

^R^^^  yj         is  to  be  measured.     Let  CD  be  an  up- 

^^^^K>         y/^'  right  pole,   15  feet  above  ground.     Let 

^HHK-rv  y/^  E  be  the  mark  in  the  tree  for  the  height 

^^^    ^y^  of  your  eye,  ^\hich  suppose  to  be  5  feet; 

■K'      j/\  and  F  the  maik  in  tlie  pole,  of  the  saute 

height.     Thtn,  FD,  the  part  of  the  pole 
above  the  mark,   will  be   1     feet.     Let 
G  be  the  place  of  your  eye,  which  is  in 
a  range  with  D   and  B,  the  top  of  tlie 
pole  and  the  top  of  the  tree  ;  and    let  II  be  the  s^t^tion,  or 
place  where  you  >tand.     Let  tlie  distance  from  H  to  C  mea- 
sure   5  feet,  and  from  H  to  A  45  tV et.     Then, 
I      ,.        HC     :     F!)     :  :     HA 
I  15     :       10     :  :       45     : 

\   /the  tree  above  the  mark;  and 
!    height  of  the  tree. 

J\^ote.  ff  the  tree  is  not  perpendicular,   but  leans,   let  ilie  pole  be  placed 
paral'el  to  it.  and  tiie  same  process  will  ^ive  its  1  ngtb, 

^H^  Problem  7.  To  i/ieasitre  the  breadth  of  a  river, 

^^E'  FULE. 

^H  B         Take  any  station,  A,  on  one  side;  and  se- 

^H;  A         lect  any  object,  B,  on  the  other  side^   oppo- 

^^'  ^  site  to  A.     Measure  back  any  distance,-  to 

C,   in  a  range  with  BA,   and  note  the  dis- 
tant e.     At  any  distance    from    A,   take  a 
E/    _        j         point  D,  and  also  such  a  point,  E,   as  shdll 
be  in  a  range  with  BI),  and  so  that  EC  shall 
be  parallel  te  DA.     From  D,  make  DF  parallel   to  AC,  and 
it  will  be  of  the  same  length ;  and   measure   EF   and  DA. 
Then  say,  EF  :  FD  : :  DA  :  AB,  which  is  the  breadth  of 
the  river  required. 


EB,  that  is, 

>  ,  which  is  the  height  of 

30-f-5    is  35  feet,  the  true 


Da 


148  Mensuration, 

Example.  Let  AC  be  20  rods,  and  of  course  FD  is  20  rods. 
Let  EF  be  12  rods,  and  DA  18  rods.     Then, 
EF     :     Fi)     :  :     DA     :     AB ;  thiit  is, 
12     :      20     :  :       18     :       SO  rods;       which    is   the 
breadth  of  the  river. 

Definitions.  A  solid  is  that  which  has  length,  breadth, 
and  thickness. 

A  euhe  is  a  solid  bounded  by  six  equal  squares. 

Problem  8.   Tujtnd  the  solid  content  of  a  laad  of  wood, 

HuLE.  Multiply  the  length  by  the  breadth,  and  that  pro- 
duct by  the  height,  and  the  last  product  will  be  the  content. 

J^Tote  The  solid  content  of  a  stick  of  squ;ired  timber,  the  height  and  breadtk 
of  which  do  not  vaiy  froni'oue  end  to  the  other,  is  found  in  he  satno  manner. 
But  if  it  tapers  regularly  throug)i»«ut,  it  is  the  frustum  of  a  p)ramiii  ;  and 
its  solid  content  may  be  found  by  the  following 

Rule.  Add  into  one  sum  the  areas  of  the  two  ends,  and  the 
square  root  of  th  ir  product ;  and  take  one  third  of  that  sum 
for  the  mean  area,  which  being  multiplied  by  the  length  of 
the  frustum,  will  give  the  solidity. 

Example.  A  stick  of  squared  timber  measures  as  follows: 
At  the  butt  end,  14  inches  by  U'  ;  at  the  small  end,  10  inches 
by  8  ;  and  the  length  is  20  feet.     What  is  the  solid  content  ? 

Operation,  14 x  12=168  inches  is  the  area  of  the  butt  end, 
and  8x10=80  inches  is  the  area  of  the  small  end.  I'38x80 
58=1.3440,  and  the  square  root  of  that  is  1 1 5*93.  And  t^8-f- 
80-1- 1 1 5*9  )  is  363-93,  of  which  one  third  is  2  '31,  which  is 
the  mean  area  in  inches.  This  multiplied  by  the  length,  240 
inches,  gives  29 1 14-4  inches,  or  16  feet,  1466*4  inches,  for 
the  solid  content. 

Pkoblem  9.  To  find  the  superficial  content  of  a  right 
cone. 

Rule.  Multiply  the  circumference  of  the  base  by  the 
slant  height,  and  to  half  the  product  add  the  area  of  the  base, 
and  it  will  give  the  superficial  content. 

JVote  The  superficial  onttnt  of  a  ri}2:ht  pyramid,  is  foiind  in  the  same 
jnanner,  the  slant  height  h«*ing  measured  oo  a  line  let  down  from  the  vertex 
})erpttndicularly  upon  the  base  of  the  triangle  which  forms  one  side  of  the 
pyVamid. 

Pro B lem  1 0.  To  jind  the  solid  content  of  a  right  pyram id, 
or  cone. 

Rule,  Find  the  area  of  the  base,  and  multiply  that  area  by 
the  perpendicular  height,  and  one  third  of  the  product  will 
be  the  solid  content. 

Problem  li.  To  find  the  relaiive  proporti^xn  of  similar 
$$lid^ 


II 


Mensuration,  149 


Rule.  Similar  solids  are  to  each  other  as  the  cubes  of  their 
similar  dimensions. 

Example.  If  a  cone,  the  diameter  of  whose  base  is  3  feet, 
contains  100  solid  fe^t,  how  many  solid  feet  Mill  a  similar 
cone  contain,  the  diameter  of  whose  base  is  6  feet  ? 

Operation,  The  cube  of  3  is  27,  and  the  cube  of  6  is  £16  ; 
therefore,  27  :  216  ::  100  :  800  feet,  Jns, 

ProblExM  12.  Tojind  the  solid  content  of  a  cylinder. 

Rule.  Multiply  the  area  of  the  base  by  the  height,  and 
the  product  will  be  the  solidity. 

J^''ote.  A  stick  of  round  timber,  ot'tlie  same  diameter  throughowt,  is  a  cy- 
linder. If  the  stick  tapers  then  it  is  ihe  frustum  of  a  cone  ;  and  its  soliri 
•ontent  may  be  found  in  the  same  mantier  as  the  solid  content  of  the  frus- 
tum f)f  a  pyramid      See  note  to  prohkni  8. 

Problem  13.   Tojind  the  surf  ace  af  a  globe  or  sphere. 
Rule.  Multiply  the  circunjference  by  the  diameter,  and 
.the  product  will  be  the  superficial  content.  ^  ** 

Problem  14.  To  find  the  solidity  of  a  globe  or  sphere. 
Rule.  Multiply  the  surfaoe  by  the  diameter,  and  one  sixth 
of  the  product  will  be  the  solid  content. 

Problem  15.  To  find  the  capacity  of  a  cask  of  the  usual 
form. 

Rule.  Add  into  one  sum  39  times  the  square  of  the  bung 
diameter  in  inches,  24  times  the  square  of  the  head  diameter, 
and  26  times  the  product  of  those  diameters  ;  multiply  that 
sum  by  the  length  of  the  cask,  and  that  product  by  '00034  5 
and  the  last  product  divided  by  '?,  will  give  the  content  ia 
wine  gallons,  and  by  1 1,  in  ale  gallons. 

Example.  What  is  the  capacity  of  a  cask,  of  which  the 
head  diameter  is  27  inches,  the  bung  diameter  33  inches,  and 
the  length  36  inches  ? 

Operation. 
33x33  =  1089,  and  1089x39=42471 
27x27=  729,  and    729x24=  7496 
33x^7=  891,  and    89lx2*=23166 


83133 

And  83133x36=i?992788,  and  ^9927fc8x-0C0S4=l0I7 
•54792,  and   1017-5479 '^--9  =  1 13'060S8. 

So,  the  answer  is  1 13'0"088  gallons,  wine  measure. 

Problem  16.   To  find  the  tonnage  of  a  ship. 

Rule.  Multiply  the  length  of  the  keel  in  feet,  by  the 
breadth  of  the  beam,  and  that  product  by  half  the  breadth  of 
the  beam  ;  and  divide  the  last  product  by  95  ;  the  quotient 
will  be  the  number  of  tons. 

N2 


lot  Mensiiratioiu 

Problem  17.  To  find  the  solid  content  of  an  irregular 
body. 

Rule.  Put  it  into  any  cylindrical  or  cubical  vessel,  and 
fill  the  vessel  with  water,  sand,  or  any  other  convenient  sub- 
stance. Then  take  out  the  body,  and  measure  the  space  left 
empty  in  the  vessel  by  its  removal,  according  to  the  preceding 
rules. 

J\^ote.  It  was  by  the  help  of  this  rwle  that.  Archimedes  discovered  the 
cheat  that  was  practised  upon  Hiero,  km^  ofSyiacuse,  respecting-  his  crown. 
He  hud  dii-ected  a  crown  of  pure  gold  to  oe  tr^aoe  ;  but  suspected  the  woik- 
man  Ijad  mixed  alloy  with  it.  He  therefore  requested  Archimedes  to  ascer- 
taii)  the  fact,  without  injuring  the  c.own.  Archimi-des  took  a  muss  of  pure 
goUl,  and  another  of  alley,  eacli  equal  i  -  weight  to  the  crown  ^  and  puiting 
e^ch  separately  ii.t<:  a  vessel  fil'ed  with  waer,  observed  the  quantity  of  water 
cxpi-lled  by  each  ;  from  which  he  ascei'taii-ed  their  respective  bulks,  and 
the  qua;. lit)  of  gold  and  alioy  which  w^^re  mix<  d  in  the  crown. 

Supp<'^e  the  weight  of  the  crown  and  of  each  nr.-ss  to  be  10/^,9.  ;  and  \].itt, 
on  being  put  into  watei-,  the  allo}  expelled  -O'i/i?) ,  the  tjoltv,  '5'Zlb  ^  arWJ  the 
compound,  -Gi/o.  Then,  by  case  3d  of  ailigation,  the  proportion  of  gold  and 
liltoy  may  be  found,  as  foiiows  . 

•92.    -12  of  alloy,  }    =i*40  ;  but  there  ought  robe 
•52^    -28  of  gold,  >    !0^6.;  therefore, 
::  12  :  SlL  of  alloy,  and 


28  :  7^&.  ofgold,  S    ^^'^' 

Problem  18.  To  find  what  weight  may  be  raised  by  any 
poiver  with  a  lever. 

Rule.  As  the  distance  between  the  weight  and  the  prop, 
is  to  the  distance  between  the  prop  and  the  power,  so  is  the 
power  to  the  weight  it  will  balance. 

Example,  if  a  man,  weighing  ioOlhs.  rest  on  the  end  of  a 
lever  lO  feet  from  the  prop,  what  weight  will  he  balance  at 
the  other  end  of  it,  1  foot  from  the  prop  ? 
1  :   10  ::  l/>0  :  l5  '0  Lbs.  Ans. 

sJVote  No  allowance  is  here  madf;  for  the  weight  of  the  lever,  whicli  ought 
to  be  done  in  order  t©  obtain  the  exact  answer 

Problem  \ 9-  To  find  what  weight  may  be  raised  by  any 
power,  with  the  wheel  and  axle. 

Rule.  As  the  diameter  of  the  axle,  is  to  the  diameter  of 
tlie  wheel,  so  is  the  power  to  the  weight  it  will  balance. 

Problem  ^.0.  To  find  what  weight  may  be  raised  by  any 
poiver,  with  a  screw. 

Rule.  As  the  distance  between  the  threads  of  the  screwy 
is  to  the  circumference  described  by  the  end  of  the  lever,  so 
is  the  power  to  the  weight. 

JS'^Qte.  One  third  of  the  eifect  ©i'this  macliine  skould  be  abated  for  frictiw. 


Mensuration.  151 

Problem  21.  To  find  what  weight  may  he  raised  by  any 
jpowevt  with  a  pulley. 

Rule.  If  the  pulley  is  fixed,  the  power  and  the  weight  arc 
equal  ;  but  if  the  pulley  is  movi?ble,  as  l  to  the  number  of 
ropes,  so  is  the  power  to  the  weight. 

Problem  2:2.  To  measure  any  height,  by  the  time  a  heavy 
body  would  fall  from  it  to  the  gTGiind. 

Rule.  In  the  tirst  sec  jnd,  it  would  fall  l6  feet,  and  in  the 
next  48,  and  so  on,  with  a  velocity  uniforn.ly  increasing. 
Therefore,  as  I  is  to  i  6,  so  is  the  scjuure  of  the  number  of 
seconds,  to  the  number  of  feet  through  which  the  borly  fulls. 

Example,  if  a  bullet  falls   from  the  t)j»  ot  a  steeple  in  S 
seconds  of  time,  what  is  the  height  of  the   sleepier 
1     :     16     :  :     9     :     J44  feet,  Ans. 

Question,  How  deep  is  a  chasm,  into  which,  if  you  drop  a 
stone,  it  will  be  10  seconds  before  you  hear  it  strike  the  bot- 
tom ? 

Operation,  Part  of  the  time  is  occupied  by  the  f^illitig  of 
the  stone,  and  part  by  the  return  of  tlie  sound  after  the  stune 
strikes  the  bottom.  To  ascertain  which,  I  work  by  double 
position,  as  follows  : 

First,  I  suppose  the  depth  to  be  1250  feet.  Then,  by  the 
above  rule,  l6, :  1  ::  12i0  :  the  square  of  the  time  occu* 
pied  by  the  stone's  falling ;  which  proportion  being  worked 
out,  gives  r?>*l25  for  the  square  of  the  time ;  and  the  square 
root  being  extracted,  is  8*838-f-  seconds,  for  the  time  of  fall- 
ing. This  being  tak^m  from  10,  the  whole  time,  leaves  1*  i  (  2-f 
for  the  sound  to  return.  And  as  sound  flies  1-12  feet  per 
second,  I  multiply  1*1 62 —  by  1'a%  and  it  gives  1327*004 — 
feet,  tbr  the  distance  it  returns.  But  it  ought  to  be  *»nly 
1250,  by  the  supposition  ;  consequently  the  first  error  is 
77-004—  too  great. 

Secondly,  1  suppose  the  depth  to  be  1260  feet,  and  proceed 
in  the  same  manner,  and  find  7S'75t  for  the  square  of 
the  time  of  the  fall ;  and  8-874+,  its  square  root,  is  the 
time;  which  taken  from  10,  leaves  1-  25-f,  for  the  time 
of  the  return  of  the  sound;  and  this  multiplied  by  1142, 
gives  1284-75+, for  the  distance  it  leturns,  which  ought,  by 
the  supposition,  to  be  only  1260 ;  so,  the  second  error  is 
24-75+  too  great. 

Next,  1250x24-76  is  30957-5,  and  1260x77-004  is  97025-04, 
and  their  difterence  is  66087-54,  which  being  divided  by 
52-254,  the  difference  of  the  errors,  gives  1264*73+  fett,  for 
the  answer  or  depth  cf  the  chasm. 


152  jfppendix  to  JSTumeration. 

"^l??  I J  ■§  S  f  5                                          •sp^apimii  t^  I  To  1 1 

«   cr  '2  ^-^    COO                                                              •SpUUSUOIl^  ^  3    53    ^    2 

.gcc-«QiC:ra3^                             .spTivjsnoii^  |()  sua}  ^  "^  *  ?  "^  "o 


i  ;u  «.^ 


^•^  'spuBSUOmjo    piinq 


^ 


tj=;  c5 ' 


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c  >  '*'  ^  '-ti  2  •*-  ■=  -^                           *I!^^'  J<^  spa.ipunij  Oi  •^z.-Xt^ 

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ji5:J35^-o^'3  'if.ipi?' bjo  spn«sno«ji  C5  ©-^  S^?^"^ 

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w'f  :;  <i^  ■=  ^^    --5  '                 •jiminb  p  sao:)  co  |r^  § 'H  ^' w 

•£S:5-S£^^-|:;^  -n^^ini)  JO  sp^jpimii  ^5-?  ^  I -f  1 1  -g 

« J  c'=  sTi^  o  %  ^  ^^|nuTi.bjospuT?s:.oq)  ^  t3  ^  §"3  ^  I 

gccj'*i==SQ"5  •pimnb  jo    noqi  )o  su^.;  ^  «  "  *^  ^  -c  § 

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«mii5:2-uE:':^  •jl^^ojospn^isnoq}  'O        |  i>  ^-1  « f^ 


?      "^  i  2  «•  nl  S  -inoojo  -nori}  |0  su,a  -?        ^  i  1 1  -t^  'i 

.  -^        .  -  c«  -  w  'jpoojo  -nom  j«>   punq  ^        '*'~~i       a^ 

5S  'SiiotutuQ^i  Q^ 


^?2^«.2i5i        ''•^'^•*'' '"""? •^"  P"""^  -  ^ ;:?^ ^ -^ 


! 

I 


PART  IT. 


J\1ote.  The  questions  which  follow,  are  intended  for  the  practice  of  thofie 
who  are  put  suing  the  studv  of  Arithmetic.  As  soon  as  the  leafier  has  been 
sufficiently  extrcised  in  the  questions  in  Pari  I.  under  simple  additio.  ,  in  ihe 
mannei* there  directedi  he  should  be  put  upon  performing  these;  cure  beij>^ 
taken  however,  that  he  should  proceed  in  learning  ihe  rules  in  Part  I.  as 
fast  as  he  proceeds  in  performing  the  quesiious  in  this  Ihe  first  600  ques- 
tions go  through  all  the  rules  in  the  book,  being  qutstio^s  ot  ihe  most  simple 
form.  'l"he  Instructor  of  the  school  should  fur-  ish  himself  with  a  Key,  con- 
taining the  answer  to  each  qu'  stion  annexed  to  its  number,  so  that  wb  n  a 
question  is  performed  by  any  scholar,  he  can  see  at  f»uce  whether  the  a;  sh  r 
is  right  If  not  right,  the  scholar  should  be  s  t  to  wotk  it  out  'gain,  and  not 
be  told  how,  till  he  hss  mad^^^  sufficient  trial  of  his  own  skill  If  the  school  is 
large,  and  the  txamination  of  questions  should  be  tr«u^  les-  me  to  the  Instruc* 
tor,  Monitors  may  be  appointed  to  do  it ;  ai^d  the  dfff.  rent  parts  of  the  Key 
may  be  put  into  their  hands  for  that  pui-pose  I'o  prevent  scholars  front 
copying  their  answers  from  ^^ach  other  they  should  be  pt  evented  riom  kecp- 
ingthetii,  and  be  directed  to  show  the  work  of  each  question  to  the  Instructor 
or  Monitork  ^should  th«  Instructor  think  aecessar>,  he  can  diiect  a  scholar 
to  go  over  the  first  600  q«estions  a  second  time,  before  !ie  iroceed  furtiier. 
The  questions  from  No.  601,  to  the  end,  are  on  all  the  rules  promiscuously, 
s©me  of  the  ino»t  difficult  being  placed  towards  the  last. 


QUESTIONS. 

No.  1.  The  Old  Testament  is  divided  in  the  following 
manner  :  The  Pentatf^uch,  containing  five  books  ;  other  his- 
torical books,  12  ;  the  Hagiographa,  5  ;  and  the  Prophets,  i7. 
How  many  books  in  all  ? 

2.  The  New  Testament  contains,  the  Gospels,  4  books  ; 
Acts  of  the  Apostks,  1  ;  Epistles  of  Paul,  14  ;  of  thi-  other 
Apostles,  7  ;  and  the  Revelation,  1.     How  many  in  ail  ? 

3.  How  many  books  in  the  whole  Bible  ? 

4.  In  the  Pentateuch,  there  are  i  87  chapters;  other  histo- 
rical books,  ^^49;  Hagiographa,  243  ;  Prophets,  250.  How 
many  chapters  in  the  Old  Testament  ? 

5.  The  Gospels  contain  ^9  chapters ;  Acts,  ?8  ;  Epistles, 
121  ;  Revelation,  22.  How  many  chapters  in  the  New  Tes- 
tament ? 

6.  How  many  chapters  in  the  whole  Bible  ? 

7.  In  the  year  1821,  the  number  of  onlained  missionaries 
employed  among  the  heathen,  was  as  follows  :  By  the  Society 
in  England  for  propagating  the  Gospel,  1  ;  Soc.  for  promoting 
Christian  knowledge,  3  ;  Danish  Mission  College,  2  ;  Mo- 
ravians, 68  ;  English  Methodist  Missionary  boc.  74 ;  English 


154  ^vesfions. 

Baptist  Miss.  Soc.  28  ;  London  Miss.  Soc.  85  ;  Scotch  Missi, 
Soc.  7 ;  Eng;lish  Church  Miss.  Soc.  32  ;  London  Jews'  8oc. 
6  ;  En^ish  Soc.  for  conversion  of  Negro  Sldves,  6  ;  Amerscaa 
Board  for  Foreign  Missions,  24  ;  American  Baptist  Do.  t ; 
United  For.  Miss.  Soc.  7  ;  American  Methodist  Miss.  Soc.  1. 
How  many  in  all  ? 

8.  The  number  of  graduates  at  the  Colleges  in  New-Eng- 
land, in  the  year  U>}0,  was  as  follows:  Harvard,  6i  ;  Yale, 
54;  Dartmouth,  26  ;  Williams,  28;  Brown,  0  ;  Burlington, 
17  ;  Middlehury,  9  ;  Bowdoin,  ir..     How  many  in  all? 

9.  From  the  creation  to  the  flood,  was  165  years;  from 
tliat  to  the  call  of  Abraham,  427  ;  from  that  to  the  departure  of 
the  Israeiitesout  of  Egypt,  480;  from  that  to  the  building  of 
the  temple,  479  ;  from  that  to  the  founding  of  Rome,  266  ; 
from  that  to  the  birth  of  Christ,  74-8  ;  and  from  that  to  tl\e 
eonnnencement  of  the  Christian  era,  4  ;  and  from  that  to  the 
pre?ent  year,  '  h22.     How  long  since  the  creation  ? 

10.  The  follow  ng  sums  were  subscribed  to  the  American 
Bible  Society,  in  a  few  days  after  its  formation,  to  wit :  By 
Elias  Boudinot,  ten  thousand  dollars  ;  John  Langdon,  400  ; 
Robert  Oliver,  300-;  Matthew  Clarkson,  .0;  Ann  ISancker, 
10  . ;  and  John  Jay,  1 50.  How  much  was  subscribed  by  these 
six  persons  ? 

1 1.  The  founders  of  the  Antlover  Theological  Semii^ary 
had,  in  the  year  i  820,  given  to  it  as  follows  :  Samuel  Abbot, 
on^  hundred  thousand  dollars;  William  Bartlet,  90000; 
Mrs  Norri*^,  30(  0«. ;  Moses  Brown,  .S:-  00  ;.  William  Phdips 
&  Son,  15000  ;  and  J^.hn  Norris,  lOOOO.     How  ^^ach  in  all  ? 

J  2.  The  army* of  Bonaparte,  when  he  invudc  Ku?.sia,  con- 
sisted of  25000(*  French,  ^0  00  Pole.s,  ^0  00  Haxof^s,  M^  00 
Austriaus.  3^  u-  0  Bavarians,  2'20  0  Prussians,  .;.:•:  «^'^  W^st- 
phalians,  8 '00  Wirtembergers,  19  0^*  subjects  of  ihe  smaller 
German  Princes,  20000  Neapolitans  an<l  IfaiiRns,  10(00 
Swiss»and  4  0    Spanish  and  Portug'^ese.     How  many  in  all? 

13.  The  population  of  the  world  is  estimated  as  follows: 
Europe,  I  "937  4000;  Asia,  :M)0  millions  ;  Africa,  V894j()0  ; 
Nor^h  \  n  pnca,228  '■  .j00l>  ;  South  America,  1 5  millions.  How 
many  in  all  ?  -« 

1  i  lie  number  of  nomiiaal  Chriltians  is  estimated  a-,  fol- 
lows: [n  Europe,  175f>65  00  ;  America,  ?^8  .>000 ;  Asia, 
300000    ;  Africa,  40!  0000.     How  many  inall  ? 

1 5.  The  number  of  Mahometans  is  estlm'd<<Hl  as  follows  : 
In  Europe,  4000000;  Asia,  57000000;  Africa,  30000000. 
How  many  in  all  ? 


^uestiom.  155 

1 6.  The  number  of  Jews  has  been  lately  estimated  as  fol- 
lows: In  Poland,  one  million;  Russia,  20000  ;  Germany, 
500r*00;  Holland  and  Netherlands,  8000a ;  Sweden  and 
Denmark,  oOOU ;  France,  3  000 ;  England,  50000;  Italy, 
20000'  ;  Spain  and  Portugal,  lOOOB ;  United  States,  300o ; 
Mahometan  states,  4  millions ;  Persia,  and  the  rest  of  Asia, 
50000^.     How  many  in  all  ? 

17.  The  Jews  were  dispersed  from  the  taking  of  Jerusalem 
by  Titus,  in  the  year  7  ,  and  the  New  ^  estament  was  pub- 
lished in  Hebrew,  by  the  London  Society  for  the  conversion 
of  the  Jews,  in  the  year    8  w  ;  how  long  between  ? 

18.  The  London  Religious  Tract  Society,  from  its  formation 
in  the  year  79  ,  in  s-ix  years,  issued  two  millions  of  tracts; 
and  in  their  thirteenth  year,  issued  2  6  00:  how  many 
more  did  they  issue  that  year,  th;in  m  the  first  six  years  ? 

li^  In  thirteen  years,  they  had  issued  145  OlO(  ;  how 
many  of  these  were  issued  during  the  second  six  years? 

2  .  The  l^ritish  and  Foreign  Bib.e  Society,  in  the  first  se- 
venteen years,  had  distributed,  or  assisted  in  distributing 
5  45  3  Bibles  and  Testaments,  of  which  3270!  6.  were  from 
their  own  depositories ;  how  many  were  thv^  rest  ? 

2:.  The  American  Bible  Society  was  formed  in  the  year 
1816,  which  wa^^  i 2  years  after  the  formation  of  the  British 
and  Foreign  Bible  Society,  and  that  was  13  years  after  the 
formation  of  the  English  Baptist  Missionary  Society,  and  that 
was  5^^  years  after  the  comuiencemeht  of  the  Moravian  mis- 
sions, and  that  was  17  years  after  the  commencement  of  the 
Danibh  mission  to  Ti anquebar ;  in  what  years  did  each  of 
these  take  place  ? 

2v.  In  its  first  four  years,  the  American  Bible  Society  had 
issued  74674.  Bibles,  and  I  4  )  Testaments ;  how  many  more 
of  the  foi  mer,  than  ot  the  latter? 

23  In  the  year  8  9,  Leander  Von  Ess,  Roman  Catholic 
professor  of  divinity  at  Marburg,  had  distributed  among  his 
Catholic  brethren,  v-.39<  7r>  copies  ot  the  New 'le^tament,  of 
which  10o434  were  during  the  last  year ;  how  many  before 
that  r 

24.  The  Connecticut  Bible  Society  distributed,  in  their 
first  4  yt*ars,  7644  Bibles,  of  which  2341  were  in  the  fourth 
year ;  how  many  in  the  other  three  r 

2^.  In  eight  years,  they  had  distributed  18056.  Bibles; 
kow  many  m  the  second  four  years  ? 


156  ^uesthns, 

26.  The  first  Sabbath  school  was  established  at  Gloucc^terj 
England,  by  Robert  Raikcs,  in  the  year  7S'^.  ;  and  Robert 
M<»rrjsoa,  who  receved  his  first  reliuious  impressions  at  a 
Sabbath  school,  and  afierwirds  became  a  tnissiouary,  finish- 
ed kis  translation  of  the  Bible  into  Chinese  in  the  year  I8i9. 
How  long  between  ? 

*^7.  The  Russian  Bible  Society  was  formed  in  1813,  and 
that  was  36  year-,  after  the  Emperor  Alexander  was  born,  and 
that  was  5t  years  after  Russia  bscame  an  empire,  and  that 
was  740  years  after  Christianity  was  introduced  into  Russia  ; 
in  what  years  did  each  of  these  take  place  ? 

28.  In  one  month  of  the  year  181 P,  the  number  of  negro 
slares  brought  into  Cuba,  was  1 728  ;  how  many  would  thai  be 
in  a  year,  at  the  same  rate  ^ 

29.  In  consequence  of  the  numbers  that  were  brought  in, 
the  price  had  fallen  to  450  dollai's  each;  what  would  the 
number  for  a  year  amount  to  ? 

3;-.  The  number  of  slaves  transported  from  Western 
Africa  for  25  years,  ending  in  18  i  9,  was  stnted  to  be  such  as 
w^ould  average  sixty  thousand  a  year ;  what  was  the  whole 
number  in  that  time? 

31.  What  would  be  the  whole  value,  at  the  above  price  in 
Cuba.?^ 

3^.  In  1815,  the  sum  received  by  the  two  London  theatres, 
was  *?tated  nt  L>00  sterling  a  night ;  suppose  this  to  be  con- 
tinued 5  months,  25  ni^ihts  in  a  month,  what  would  be  the 
auKunt  r 

S3.  In  December,  1814,  the  nine  Paris  theatres  were  stated 
to  have  received  L\  Hi^oo  sterling  ;  if  this  continued  5  months, 
"what  was  the  amount  ? 

34.  In  18  1 8,  the  London  Hibernian  Society  supported  480 
charity  schools  in  Ireland,  which  averaged  about  98  scholars 
each  ;  how  many  poor  children  were  receiving  an  education 
from  this  charity  r 

3.5.  The  same  year^  the  T^ondon  Sabbath  school  society  for 
Ireland,  assisted  634  schools,  which  averaged  about  1  2  scho- 
lars each  ;  how  many  poor  children  were  receiving  aid  from 
this  charity  } 

r6  A  ujan  who  depended  on  his  dady  labor  to. support 
himself  and  family,  appropriated  the  earnings  of  one  half  day 
each  month  tocharifable  purposes,  and  in  one  year  the  amount 
wah  ten  dollar-.  The  militia  rolls  of  the  United  States  con- 
tain the  n.imes  of -7o828  men.  If  each  of  these  should  "go 
a»d  do  likew  ifce,"  how  aiuch  would  be  thus  raised  annually  ? 


Questions,  157 

37.  The  Royal  Mission  Chapel,  built  bj  King  Pomarre,  in 
Otaheite,  and  dedicated  in  May,  18 1 9,  is  stated  to  be  712 
feet  long,  and  54  wide ;  how  many  square  feet  does  it  contain? 

38.  If  4  square  feet  be  allowed  to  each  person,  how  many 
persons  would  it  accommodate  ? 

39.  The  whole  number  of  schools  in  Scotland  in  1820, 
was  3556,  in  which  were  taught  1 76303  children ;  what  is  the 
average  number  for  each  school  ? 

40.  In  the  year  18)^;,  the  number  of  schools  in  Ireland, 
under  the  patronage  of  the  London  Hibernian  Society,  was 
534-,  and  the  number  of  scholars  in  them  54520 ;  how  many  to 
each  school  ? 

4t.'  In  the  year  1665,  Connecticut  contained  abount  9000 
inhabitants,  and  had  21  ministers  of  the  gospel ;  how  many 
souls  to  each  minister  ? 

42.  In  1713,  the  inhabitants  w^ere  17000,  and  the  ministers 
and  licensed  preachers  45  ;  how  many  souls  to  each  ? 

43.  The  annual  expense  to  the  inhabitants  of  Boston  for 
the  support  of  their  theatre,  was  estimated,  in  1820,  at  75000 
dollars;  how  many  missionaries  would  that  sum  support 
among  the  heathen,  at  5oo  dollars  each  ? 

44.  How  many  would  the  receipts  of  the  London  theatres 
support,  at  ^125  sterling  each  ?     (See  No.  32.) 

45.  How  many  would  the  receipts  of  the  Paris  theatres 
support,  at  the  same  rate  ?     (See  No.  33.) 

46.  The  New  England  Tract  Society' was  formed  in  the 
year  1814,  and  in  seven  years  had  published  2708000  tracts ; 
what  is  the  average  number  per  year  ? 

47.  In  the  western  part  of  Virginia,  there  was  stated  to  be, 
in  1821,  a  district  containing  l7o(X)0  souls,  and  only  8  edu- 
cated ministers  of  the  gospel ;  how  many  souls  to  each  mi- 
nister ? 

48.  The  population  of  London,  in  l81 1,  was  1039000,  and 
it  is  estimated  that  there  are  212000  strangers  constantly 
there  ;  how  many  in  all  ? 

49^.  If  half  these  attend  public  worship,  how  many  churches 
are  necessary,  each  accommodating  80    persons  ? 

50.  In  I8i6,  there  were  the  following  places  of  religious 
worship  in  London:  EpivScopal,  166;  Dissenters,  136; 
Dutch  and  German,  19;  Catholic,  13;  Jews,  6;  Quakers,  6. 
How  many  in  all  ? 

51.  How  many  more  wore  needed? 

O 


158  Questions. 

52.  By  official  returns,  it  appeared,  in  1820,  that  there 
were,  in  England  and  Wales,  37382  schools,  and  1571372 
children  taught  in  them  ;  what  is  the  average  number  to  a 
school  ? 

53.  At  the  same  time,  in  France,  1075500  children  were 
learning  to  read  and  write,  under  the  care  of  28000  masters; 
how  many  scholars  is  that  for  each  master  ? 

54.  The  following  were  the  receipts  of  the  principal  reli- 
gious charitable  societies  in  England,  in  the  year  1 820,  to  wit : 
13.  &  F.  Bible  Society,  ^89154  sterling;  Christian  Know^- 
ledge  Soc.  ^53100;  Church  Miss.  Soc.  ^31200;  London 
Miss.  Soc.  ^56174  ;  Methodist  Miss.  Soc.  =g225@0  ;  Baptist 
Miss.  Soc.  jei3200;  Soc.  for  propagating  the  Gospel,  ^13000; 
Soc.  for  conversion  of  Jews,  £  I  tr780 ;  National  Soc.  for 
Education,  L8./00;  Religious  Tract  ^oc.  jLr56l  ;  Hibernian 
Soc.  L7049;  Moravian  Missions,  LsOOO;  Navul  and  Mili- 
tary Bible  Soc.  £2348  ;  Br.  &  For,  School  Soc.  ^£034 ; 
Prayer  Book  and  Homily  Soc.  il993.     How  much  in  all  ? 

55.  The  first  missionaries  landed  at  Otaheite  in  the  year 
1797, and  idolatry  was  abolished  in  I8l5  ;  how  long  between? 

56.  The  missionaries  sailed  for  the  Sandwich  Islands  in 
the  year  1819,  which  was  2  years  after  the  establishment  of 
the  Cherokee  Mission,  and  that  was  5  years  after  the  first 
missionaries  sailed  from  America  to  India,  and  that  was  2 
years  after  the  American  Board  for  Foreign  Missions  was 
formed,  and  that  was  2  years  after  the  Theological  Seminary 
was  established  at  Andover ;  in  what  year  did  each  of  these 
take  place  ? 

57.  It  is  computed,  that  in  the  year  1813,  at  least  800000 
men  died  in  war,  and  200(00  more  were  maimed  for  life, 
and  rendered  useless ;  and  it  is  reckoned  that  the  pecuniary 
loss  to  the  public,  from  the  death  of  an  able  bodied  man,  is 
j^rO  dollars  ;  if  so,  what  is  the  whole  loss,  in  this  way,  o£ 
that  one  year  ? 

58.  it  is  computed  that  the  United  States  lost  17000  men 
in  the  late  war  with  Great  Britain  ;  what  is  the  amount  of 
that  loss,  on  the  same  principle  ? 

59.  The  English  Sabbath  School  Union,  in  the  year  1820, 
had,  in  the  schools  connected  with  it,  237584  scholars  ;  that 
of  Scotland,  S4000  ;  and  that  of  Ireland,  84174.  How  many 
in  all  ? 

60.  The  Inquis'tion  was  established  in  Spain  in  the  year 
148',  and  abolished  in  1808 ;  how  many  years  did  it  exist 
there?* 


Questions.  159 

61.  During  that  time,  32382  persons  were  burnt  alive  by 
its  order  ;  what  is  the  average  per  year? 

6:2.  During  the  same  time,  291450  persons  were  imprison- 
ed, and  their  goods  confiscated,  by  its  order ;  what  is  the 
average  per  year  ? 

63.  It  is  stated  that  the  following  sums  were  paid  by  the 
inhabitants  of  Charleston,  in  the  year  1820,  for  the  support 
of  the  poor,  to  wit:  Orphan  Asylum,  §22000;  Poor  House, 
D24000 ;  Marine  Hospital,  D6000  ;  Ladies  Benevolent  So° 
ciety,  D2000.     How  much  in  all  ? 

64.  Of  the  above  expense,  the  following  is  stated  to  be 
rendered  necessary  in  ccmsequence  of  the  intemperate  use  of 
ardent  spirits,  to  wit :  Orphan  Asylum,  D  \  4000  ;  Poor  House, 
DliJOOO;  Marine  Hospital,  D4000 ;  Ladies  Benevolent  So- 
ciety, DlOOO.  How  much  are  the  inhabitants  of  Charleston 
annually  taxed,  to  support  drunkards  and  their  families? 

65.  The  first  American  missionaries  to  Jerusalem,  sailed 
in  the  year  1819,  which  was  602  years  after  that  city  was 
taken  by  the  Turks,  and  that  was  SO  years  after  it  was  re- 
taken from  the  Crusaders  by  Saladirr,  and  that  was  88  years 
after  it  was  taken  by  the  Crusaders,  and  that  was  463  years 
after  it  was  taken  by  the  Saracens,  and  that  was  22  years 
after  it  was  taken  by  the  Pei*sians,  &  that  was  484  years  after 
it  was  rebuilt  by  Adrian,  and  that  was  60  years  .after  it  was 
destroyed  by  Titus ;  in  what  year  did  each  of  these  take 
place  ? 

66.  The  number  of  Lancasterian  schools  in  France  in  the 
year  1820,  was  1340,  containing^  154000  children;  how 
many  is  that  for  each  school  ? 

67.  The  number  of  students,  professors  of  religion,  and 
charity  scholars,  at  12  of  the  Colleges,  in  the  year  l821,  was 
stated  as  follows : 


Stud. 

prof. 

ch.  sch. 

Stud. 

prof. 

ch.sch. 

Yale, 

316 

97 

46 

Princeton, 

116 

25 

11 

H;i«-vard, 

291 

17 

15 

Bowdoin, 

101 

23 

7 

Union, 

255 

66 

32 

Middlebury, 

100 

48 

22 

Blown, 

151 

59 

18 

Hamilton, 

92 

48 

34 

Dartmouth, 

146 

65 

43 

Williams, 

83 

42 

24 

N.  Carolina, 

135 

10 

0 

Burlington, 

35 

9 

1 

How  many  students,  how  many  professors  of  religion,   and 
how  many  charity  scholars,  are  in  these  1 2  colleges  ? 

68.  In  the  year  1 8  19,  the  London  Religious  Tract  Society 
issued  5626674  tracts,  which  was  1583353  more  than  they 
had  issued  the  preceding  year ;  how  many  were  issued  in 
18li>? 


l60  Questions, 

69.  In  1820,  the  number  of  graduates  at  several  of  the  col- 
leges, was  as  follows  :  Union  65,  Harvard  56,  Yale  54,  Brown 
29,  Dartmouth  24,  Middlebury  22,  Pennsylvania  17, Hamilton 
14,  Bowdoin  1 1,  Burlington  9  ;  how  many  in  all  ? 

70.  in  1821,  as  follows :  Union  67.  Harvard  59,  Yale  67, 
Brown  ^0,  Dartmouth  17,  Middlebury  23,  Pennsylvania  35, 
Hamilton  18,  Bowdoin  21,  Burlington  5,  Columbia  30, 
Princeton  40,  Georgia  3  ;  how  many  in  all  tf 

71.  At  the  close  of  1816,  the  Connecticut  Miss.  Soc.  had 
sent  S579d  books  to  the  new  settlements,  of  which  5589  had 
been  sent  in  that  year  ;  how  many  before  ? 

72.  In  1821,  the  English  Methodist  Miss.  Soc.  had  among 
the  heathen  150  missionaries  and  assistants,  with  2700  con- 
verts under  their  care  ;  how  many  is  that  for  each  ? 

72.  In  1819,  the  Baptist  missionaries  in  India  had  under 
their  care,  92  schools  for  heathen  children  near  Serampore, 
1 1  at  Cutwa,  3  at  Moorshedabad,  and  5  at  Dacca ;  and  in 
these,  about  10000  native  children:  what  is  the  average 
number  for  each  school  ? 

74.  The  General  Assembly  of  the  Presbyterian  Church  in 
the  United  States,  was  formed  in  the  year  1787,  which  was 
144  years  after  the  Assembly  of  Divines  met  at  Westmin- 
ster, and  that  was  83  years  after  Presbyterianism  w^as  esta- 
blished in  Scotland  by  John  Knox,  and  that  was  26  years 
after  the  Reformation  commenced  in  England,  and  that  was 
1 7  years  after  the  Reformation  was  begun  in  Germany  by 
Luther,  and  that  was  157  years  after  the  opposition  to  Popery 
was  made  in  England  by  Wicklitfe,  &  that  was  754  years  after 
the  Pope  v^as  acknowledged  Universal  Bishop  by  the  Emperor 
Phocas,  and  that  was  10  years  after  Christianity  was  intro- 
duced into  England  by  Augustin,  and  that  was  164  years 
after  St.  Patrick  began  to  preach  in  Ireland,  and  that  was 
119  years  after  Christianity  was  established  in  the  Roman 
Empire  by  Constantine  ;  in  what  year  did  each  of  these  take 
place  ? 

75.  The  property  belonging  to  the  Choctaw  mission  at 
Elliot,  in  Dec.  1S20,  was  valued  as  follows  :  Sixty  acres  of 
improvements,  D900  ;  a  horse  mill,  D200  ;  shops,  tools,  and 
stock,  D600  ;  twenty-two  other  buildings,  D3000  ;  farming 
utensils,  D400 ;  seven  horses,  D420 ;  two  yoke  of  oxen, 
Dl60  ;  two  hundred  and  twenty  neat  cattle,  D1760  ;  sixty 
swine,  Dl50;  provisions,  D 1758  ;  groceries,  LS60  ;  house- 
hold furniture,  D500 ;  cloth,  D250  ;  library,  D.)20 ;  boat, 
B400 ;  fifty  thousand  brick,  D300.  What  is  the  whole  value  ? 


^uestions>  l6i 

76:  The  Danish  mission  to  Tranquebar,  fit  ate,  in  the  year 
1736,  that  in  29  years,  they  had  received  into  their  churches 
3^39  converts  from  heathenism  ;  what  was  the  average  per 
year  ? 

7/.  In  24  years  afterwards,  the  number  of  converts  added 
was  8267  ;  what  was  the  average  per  year  ? 

78.  In  1813,  the  Moravians  had  31  missionary  stations, 
as  follows  :  South  Africa,  2  ;  S.  America,  4  ;  N.  Aroer^a, 
7 ;  Greenland,  3  ;  and  the  rest  in  the  West  Indies  ;  how 
many  were  the  last  ? 

79.  At  the  same  time,  they  had  157  missionaries  and  as- 
sistants at  their  stations  ;  what  is  the  average  to  each  station? 

80.  In  1817,  the  number  of  converted  negroes  under  the 
care  of  the  Methodist  missionaries  in  the  West  Indies,  was 
stated  as  follows  :  In  Antigua,  3552  ;  St.  Christophers,  2552; 
St  Eustatius,  513  ;  vSt.  Vincents,  2760  ;  Bahamas,  584  ;  St. 
Barfs,  447  ;  Bermuda,  62  ;  Dominica,  638 ;  Grenada,  171  ; 
Nevis,  1183  ;  Trinidad,  267 ;  Tortola  and  Virgin  Islands, 
1664;  Jamaica,  4126;  Barbadoes,  44:  Tobago,  140,  How 
many  in  all  ? 

81.  These  were  under  the  care  of  38  missionaries  and  as- 
sistants ;  what  is  the  average  to  each  ? 

82.  The  number  baptized  by  the  Baptist  missionaries  in 
India  in  the  year  1814,  was  129;  and  the  whole  numbei* 
from  the  commencement  of  their  mission,  765  ;  how  many 
before  that  year  ? 

83.  The  number  of  ordained  missionaries  among  the  hea- 
then in  the  year  1821,  was  351,  in  the  following  countries, 
to  wit :  iVfrica,  45  ;  Isle  of  France,  2  ;  Malta,  3  ;  Ionian 
Islands,  1  ;  Polish  Jews,  5 ;  Turkey  in  Europe,  1 ;  Turkey 
in  Asia,  5  ;  Russia  in  Asia,  17 ;  China,  1  ;  India  beyond 
the  Ganges,  10  ;  India  within  the  Ganges,  78 ;  Ceylon,  26 ; 
Indian  Archipelago,  7:  Australasia,  2;  Polynesia,  17  ;  Spa- 
nish  and  Portuguese  America,  14  ;  Blacks  of  the  W^est  In- 
dies, 66;  Indians  in  the  United  States,  23;  Labrador,  19; 
and  tlie  rest  in  Greenland  ;  how  many  were  the  last  ? 

84.  If  50000  missionaries  should  be  sent  to  the  heathen^ 
?ind  1  jiOOO  heathen  should  be  allotted  to  each  missionary, 
how  many  of  them  would  be  supplied  at  that  rate  ? 

85.  It  is  estimated  that  the  annual  income  of  tlie  people  of 
the  United  States,  is  three  hundred  millions  of  dollar^ ;  ii 
one  tenth  of  this  should  be  devoted  to  the  support  of  mis- 
sipnaries,  how  many  would  it  furnish,  at  500  dollars  eack  ? 

02 


i  62  (Itustions, 

86.  In  1821,  Dr.  Carey  and  his  associates  had  translated 
and  printed  'the  whole  Bible  in  five  of  ihe  languages  of  the 
East,  the  New  Testament  in  ten  more,  and  parts  of  the  latter 
in  sixteen  more  ;  this  was  £6  years  after  they  began  the 
work,  and  that  was  84  years  after  the  Tamul  Testament  was 
published  by  Ziegenbalg,  and  that  was  £6  years  after  Elliot's 
Indian  Bible  was  |rriiited  in  America,  and  that  was  7^  years 
after  the  present  English  version  was  published  byKingJames, 
and  that  was  74  years  after  the  first  English  editioit  of  the 
Bible  was  authorised  by  Henry  the  eighth,  and  that  was  13 
years  after  the  brst  English  Testament  was  published  by 
Tyndai,  and  that  was  4  years  after  Luther  published  his 
Testament  in  German,  and  that  was  78  years  after  the  first 
books  were  printed  From  metallic  types  by  Faui^t  and  others, 
and  that  was  14  years  after  prmting  with  wooden  types  was 
invented  by  Lauren tius  in  Holland,  and  that  was  70  years 
after  Wicklitte  translated  the  Bible  into  English  ;  in  what 
year  did  each  of  these  take  jilace  ? 

87.  In  1820,  the  number  ot  Christian  pilgrims  to  Jerusalem 
was  as  follows:  Greeks,  i600;  Armenians,  i  SOO  ;  Copts, 
150;  Roman  Catholics,  50;  Abyssinians,  1;  Syrians,  30. 
How  many  in  all  ? 

88.  In  1810,  the  English  Soc.  for  promoting  Christian 
Knowledge,  distributed  10-224  bibles,  16^^42  testaments  and 
psalters,  20555  prayer  books,  20908  other  bound  books,  and 
145123  tracts  ;  how  many  in  all  ? 

89.  In  1814,  asfollovv*s:  2676^  bibles,  48018  testaments 
and  psalters,  65492  prayer  books,  5  1525  other  bound  books, 
and  6  3501  tracts ;  how  many  in  all  ? 

9c.  How  m^  ny  more  in  the  last  year,  than  in  the  otiier  ? 

91.  The  American  Bible  Soc.  in  18^0,  (their  5th  year,) 
issued  29{  00  bibles,  and  30000  testaments  ;  and  the  total  of 
copies  of  the  whole  bible,  or  parts  of  it,  issued  by  them,  was 
S3   552  ;  how  m;iny  in  the  first  four  years  ? 

9S.  The  Hegira,  or  flight  of  Mahomet  from  Mecca  to 
Medina,  was  in  the  year  622  ;  and  Constantinople  was  taken 
by  the  Turks  in  1453  ;  in  what  year  of  the  Hegira  was  ic  ? 

93.  The  New-England  Tract  Society  was  establi^hed  in 
181.S,  and  in  1821  had  published  2708000  tracts  ;  how  many 
is  that  for  each  year  ? 

94.  In  i8iO,  the  Maine  Miss.  Soc.  received  1081  dollars 
and  38  cents  ;  how  many  mills  is  that? 

95.  In  18  8,  the  receipts  of  the  English  Christian  Know- 
fedge  Soc.  were  £\S9%3  ..  9  ..  5  ;  bow  many  farthings  is  that? 


Qiiestions,  1 6S 

In  1815,  the  amonnt  was  £50226  ..  lo ..  1 ;  how  manj 
pence  is  that  ? 

9r.  In  1817,  the  amount  was  56885012  farthings ;  how 
many  pounds  is  that  ? 

98.  In  1819,  the  receipts  of  the  London  Jews  Soc.  were 
107 i:. 961.'  farthings  ;  how  many  pounds  is  that? 

99.  How  many  grains  in  367  lbs.  Troy  ? 

100.  How  many  drams  in  3  tons  ? 

101.  How  many  grains  in  45  lbs.  Apothecaries'  wt.  ? 

102.  hi  50  miles,  how  many  barley  corns  ? 

103.  How  many  lbs.  Apothecaries'  wt.  in  13337791  grains? 

104.  In  356  yds.  how  many  nails  ? 

105.  In  18  lbs.  how  many  scruples? 
lOt^.  In  64960  lbs.  how  many  tons? 

107.  in  4 1 60  poles,  how  many  acres  ? 

108.  How  many  pints  in  21  hhds.  wine  measure  ? 

109.  In  4976  pints,  how  many  bushels  ? 

110.  In  2279772  barley  corns, how  many  miles? 

111.  How  many  yds.  in  105ii  nails  ? 

112.  How  many  tons  in  20563/12  drams? 

113.  In  2  401600  seconds,  how  many  weeks  ? 

114.  In  5  hhds.  wine  measure,  how  many  gills? 

115.  How  many  poles  in  456  acres  ? 

116.  In  ^13096  grains,  how  many  lbs.  Troy? 
1  i7.  How  many  pints  in  786  bushels  ? 

118.  In  363  days,  how  many  seconds? 

119.  In  14  tons,  how  many  lbs.  ? 

120.  How  many  miles  in  34665840  inches? 

121.  In  319  nails,  how  many  yds,  ? 

122.  How  many  ounces  in  5  tons  ? 

123.  How  many  lbs.  Troy  in  245678  grains  ? 

124.  The  expenditures  of  the  Charitable  Soe.  of  Hillsbo- 
rough County,  N.  H.  for  the  year  1818,  were  as  as  follows  : 
For  bibles,  0^96-77;  domestic  missions,  D34*20  ;  foreign 
missions,  1)126*86  ;  education  of  pious  youth,  Dft09'34  :  how 
much  in  al*  ? 

}S5.  The  General  Committee  of  the  Moravians,  received 
for  their  several  missions,  in  1818,  as  follows :  Collections 
from  congregations  and  friend?,  ^1545..  2..  10  steiling; 
benefactions,  chiefly  in  England  &  Scotland,  ^4035  ..  10  ..  8; 
legacie?,  Lf.SS  ..  13  ..  2  ;  balance  from  West  Indies,  ^240  .• 
0  ..  5  ;  gained  by  exchange,  £b,»  17 ..  6 :  how  much  in  all  ? 


1 64  (luestions. 

126.  Their  |Miyments  for  missions  were  as  follows  : — 
Greenland,  L7i2..  10..  7;  Labrador,  (besides  what  was 
supplied  from  other  sources,)  ^105  ..  5  ..  11  ;  N.  American 
Indians,  ^218..  4..  4;  W.  Indies,  jL288l  ,.  9  ..  2  ;  South 
America,  ^190..  10..  11  ;  S.  Africa,  X1124..  12..  2  :  how 
much  in  all  ? 

127.  Their  other  expenses  were  as  follows  :  Pensions  to 
superannuated  missionaries,  ^748  ..  1 1 ..  2  ;  widows  of  mis- 
sionaries, ^317..  10..  S;  education  of  sixty-three  children 
of  missionaries,  L853..  15..  7;  sundries,  X.787..  14:  how 
much  in  all  ? 

128.  What  was  the  whole  amount  of  expenditure  for  that 
year  ? 

129.  In  the  year  1820,  there  was  raised  in  the  county  of 
Otsego,  N.  Y.  i  25  bushels,  4  quarts  of  corn,  on  one  acre ; 
1^20  bush.  2  pecks,  on  another  ;  1 18  bush.  4  qts.  on  another  ; 
117  bush,  on  another;  11 1  bush,  on  another;  95  bush.  4  qts. 
on  another  ;  and  90  bush.  2  pecks,  6  qts.  on  another  :  how 
much  on  seven  acres  ? 

130.  One  piece  of  cloth  contains  37  yd.  3  qr.  3  na. ;  ano- 
ther, 28  yd.  2  na. ;  another,  39  yd.  2  qr. ;  another,  9  yd.  3  na.: 
how  much  in  all  ? 

131.  Boughtof  A,  76  acres,  3  roods,  27  poles;  ofB,  26 
acres,  57  poles;  of  C,  19  acres,  3  roods;  of  D,  11  acres,  2 
roods,  17  poles  :  how  much  in  all  ? 

132.  Journeyed  on  different  days  as  follows :  36  miles,  3 
furlongs,  21  poles;  21  miles,  37 poles;  34  miles,  7  furlongs, 
28  poles ;  56  miles,  6  furlongs  ;  47  miles,  27  poles  :  how  far 
in  all? 

133.  Sold  A,  3  Cwt.  2  qr.  57  lb.  of  flour ;  B,  4  Cwt.  3  qr. 
19  lb. ;  C,  5  Cwt.  2  qr.  19  lb. ;  D,  4  Cwt.  21  lb. ;  E,  9  Cwt. 
3  qr.  18  lb.:  how  much  in  all  ? 

i34.  In  1820,  the  American  Bible  Soc.  received  D49578 
•34,  and  expended  D47759-60;  what  is  the  difterence  ? 

135.  The  American  Board  tor  Foreign  Missions  received 
D39334-51,  and  expended  D57420-93 ;  how  great  was  the 
deficiency? 

136.  The  United  Foreign  Mission  Soc.receivedD15263-3jf, 
and  expended  D 14010;  what  sum  remained  unexpended  ? 

137.  The  receipts  of  the  Ame  ican  Education  Soc.  for 
1819,  wereDl9330;  for  1820,  Dl5l4ri'80;  how  great  was 
the  falling  off? 

138.  In  1821,  its  receipts  were  D13108*97  and  its  expen- 
ditures D 10018-72;  what  is  the  difference  ? 


^uestions^  1 64 

139.  In  1819,  the  British  and  Foreign  Bible  Soc.  received 
/X9S053..  6..  7  sterling,  and  expended  jLi23.47'..  12..  3  ; 
what  was  the  excess  of  expenditure  ? 

J  40*  The  Church  Missionary  received  ZSOOOO  sterlings 
and  the  London  Miss.  Soc.  2-25406 ..  16..  4;  what  is  the 
difference  ? 

141.  In  1820,  the  London  Missionary  Society  received 
Z26174..  4..  3  sterling,  and  expended  JL27790 ..  1 7 ..  I  ; 
what  was  the  excess  of  expenditure  ? 

142.  The  London  Jews  Soc.  received  L10789  ..  18..  2 
sterling,  and  expended  X13137..  16..  1 ;  what  was  the  ex- 
cess of  expenditure  ? 

143.  Bought  2  tuns  of  wine,  and  sold  3  hhds.  25  gals.  1 
qt. ;  how  much  is  left  ? 

144.  Bought  642  lb.  9  oz.  8  gr.  of  silver,  and  sold  537  lb. 
6  oz.  10  dwt. ;  how  much  is  left  ? 

145.  Borrowed  46  Cwt.  3  qr.  16  lb.  of  hay,   and  returned 

10  Cwt.  1  qr.  26  lb. ;  how  much  remains  to  be  returned  ? 

146.  From  6  lb.  9  oz.  1  sc.  19  gr.  of  medicine,  take  5  lb. 

1 1  oz.  7  dr.  10  gr. ;  how  much  is  left  ? 

147.  If  a  man  earns  1  doll.  12^  cents  a  day,  and  should 
devote  to  the  Lord  the  earnings  of  one  day  every  month, 
what  would  be  the  amount  in  a  year  ? 

148.  If  the  fees  of  a  physician  average  D3*25  every  week 
day,  and  he  should  be  under  the  necessity  of  attending  pa- 
tients on  the  Sabbath  to  half  that  amount,  and  should  devote 
the  proceeds^  of  all  his  Sabbaths  to  Him  who  is  Lord  of  the 
Sabbath  ;  what  would  be  the  yearly  amount  ? 

149.  If  a  journeyman  mechanic  can  earn  9  cents  an  hour, 
and  perform  his  day's  work  in  9  hours,  how  much  can  he  earn 
in  a  year  for  doing  good,  by  working  one  hour  extra  each  day, 
there  being  313  working  days  in  a  year  ? 

150.  If  an  apprentice  can  earn  6  cents  an  hour,  how  much 
can  he  earn  in  a  year  for  doing  good^by  the  same  method  ? 

15 1.  If  a  young  woman  can  earn  with  her  needle,  4  cents 
an  hour,  how  much  can  she  earn  in  a  year  for  doing  good,  by 
the  same  method  ? 

152.  If  a  little  girl  can  earn  by  knitting,  5  mills  an  hour, 
how  much  can  she  earn  in  a  year  for  doing  good,  by  the 
same  method? 

is 3.  If  a  little  boy  should  raise  12  chickens  in  a  year, 
which,  when  full  grown,  should  weigh  i?  lb.  8  oz.  each,  and 
should  sell  them  for  5  cents  a  lb.  and  devote  the  avails  to  the 


166  Questions,  1 

education  of  heathen  children,  what  would  be  the  annual 
amount  ? 

154.  If  a  man  drinks  half  a  gill  of  ardent  spirits  every 
day,  how  much  is  that  in  a  year  ? " 

1^5.  If  be  makes  use  of  half  a  pint  every  day  for  himself 
and  friends,  how  much  is  that  in  a  year  ? 

156.  How  much  is  it  in  20  years,  allowing  5  leap  years  ? 

157.  What  cost  1  ;  9  cords  of  wood,  at  D^'67  a  cord  ? 

158.  What  cost  1  2  lb.  of  tea,  at  Ts.  6d.  a  lb.  ? 

159.  What  cost  96  bushels  of  rye,  at  6s.  9d.  a  bushel  ? 

160.  What  cost  11  Cwt.  of  flour,  at  Ll  ..  4  ..  6  per  Cwt.  ? 

161.  Sold  to  19  persons,  each,  17  Cwt.  3  qr.  2l  lb.  14  oz. 
15  dr. ;  how  much  in  all  ? 

162.  Bought  of  25  persons,  each,  9  lb.  10  oz.  17  dwt.  21  gr. 
of  silver;  how  much  in  all? 

163.  Mixed  16  sorts  of  medicine,  of  each  2  lb.  3  oz.  5  dr. 
1  sc.  18  gr. ;  what  is  the  weight  of  the  whole  mass  ? 

164.  Sold  to  24  persons,  each,  8  quarters,  7  bush.  3  pks. 
1  gal.  3  qt.  1  pt.  of  wheat ;  how  much  in  all  ? 

165.  Bought  13  parcels  of  wood,  each  13  cords,  127  ft. 
1727  inches  ;  how  much  in  all  ? 

166.  Bought  44  pieces  of  land,  each  16  A.  3R.  36p. ; 
how  much  in  all  ? 

167.  Sold  35  pieces  of  cloth,  each  25  yd.  3  qr.  3  na. ;  how 
much  in  all  ? 

168.  If  1  68  yds.  cost  X40  ..  12,  whatis  that  per  yd.  ? 

169.  If  55  proprietors  bought  a  tract  of  40000  acres,  what 
is  the  share  of  each  ? 

170.  If  13  persons  joined  in  purchasing  3  hhds.  of  wine, 
what  is  the  share  of  each  ? 

171.  If  2S6lb.  10  oz.  6dr.  2  sc.  18  gr.  of  medicine  be  made 
up  into  16  equal  parcels,  what  weight  will  be  in  each  ? 

172.  If  25  persons  were  joint  purchasers  of  35  Cwt.  3  qr. 
21  lb.  of  sugar,  what  is  the  share  of  each  ? 

173.  In  the  year  i819,  the  receipts  of  the  American  Edu- 
cation vSoc.  were  D{93^'0,  and  the  number  of  young  men  as- 
sisted was  161 ;  how  much  would  that  average  to  each,  if  the 
whole  had  been  distributed  ? 

174.  In  1820,  the  receipts  of  the  Western  Education  Soc. 
were  1)1755-61,  and  the  expenditures  Dl601'62  ;  whatis  the 
difference  ? 

175.  They  had  56  young  men  under  their  care ;  what 
would  the  amount  expended  be  for  each  ? 

176.  'the  receipts  of  the  Baptist  Missionary  Soc.  of  Ma  ft- 


Questions.  167 

sachusetts,  were  D2.575-68  ;  how  many  weeks  missionary 
labor  would  it  pay  for,  at  D8  a  w  *ek  ? 

177  In  the  year  ;80^  there  were  11  societies  in  the  U. 
States  for  the  support  of  missions  in  our  own  country,  and 
their  receipts  were  DlOl  0  ;  what  was  the  average  for  each? 

178.  In  18  0,  the  year  the  American  Board  for  Foreign 
Missions  was  form<^d,  the  receipts  of  the  same  societies  were 
D 10721  ;  what  was  the  average  for  each  ? 

179.  In  1818  the  receipts  of  these  same  societies  were 
D23675  ;  what  was  the  average  for  eaeh  ? 

ISO.  What  was  tht^  average  yearly  increase  of  each  so- 
ciety for  the  ^  years  before  the  Board  was  formed  ? 

IP  .  What,  for  the  8  years  after  the  Board  was  formed  ? 

182.  In  1P21,  the  American  Education  Soc.  had  under  its 
care '25f  beneficiajies,  and  distributed  among  them  D9093 ; 
what  is  the  average  for  each  ? 

183.  The  donations  to  the  Massachusetts  Miss.  Soc.  for 
18  8,  were  L356..  0..  11,  N.  Eng.  currency ;  if  that  sum 
paid  for  196  weeks  missionary  labor,  how  much  would  it  be 
per  week  ? 

184.  In  1820,  there  was  received  for  the  aid  of  charity 
students  at  the  Theoloicical  Seujinary  at  Princeton,  D2855 
•40  J,  and  the  number  of  students  was  7  i.  If  one  half  of  these 
received  aid,  how  much  would  it  be  for  each? 

185.  In  i8l5,  the  number  of  Hottentots  belonging  to  the 
settlement  at  Bethelsdorp,  was  about  1200.  The  same  year 
they  paid  in  taxes  to  government,  D3500  ;  contributed  for 
miijsions,  D  32  80  ;  collected  for  their  own  poor,  D  i  77  ;  and 
were  building  a  school  room,  and  printing  office,  70  feet  by 
80,  estimated  to  cost  at  least  D6 14-20  ;  what  does  the  whole 
amount  average  for  each  individual  ? 

186.  The  number  of  missionaries  and  assistants  employed 
by  the  American  Board  in  1820,  was  88,  and  they  had  3000 
heathen  children  under  instruction  ;  what  is  the  average  for 
each  ? 

187.  The  disbursements  for  the  several  missionary  stations 
were  0*8565  ;  how  much  is  that  for  each  missionary  and  his 
scholars  ? 

188.  The  Rev.  Joseph  Emerson  received  for  his  astrono- 
mical lectures  in  Boston,  in  1819,  D643,  and  his  expenses 
were  Dl  i  7.  If  the  remainder  was  divided  among  14  young 
ladies,  to  assist  in  qualifying  them  for  instructing  schools, 
how  much  would  it  be  for  each  ? 


168  ^utstions, 

189.  In  1820,  it  was  estimated  that  there  were  in  New- 
England,  250000  young  men,  between  15  and  35  years  of 
age.  If  70000  of  these  give  2")  cents  each  per  annum ; 
100000,  75  cents  each  ;  50000,  D2  each  ;  20000,  D5  each  ; 
and  lOOOO,  DtO  each,  to  the  American  Education  Soc,  what 
will  be  the  annual  amount  ? 

190.  How  many  young  men  would  that  assist  in  preparing 
for  the  ministry,  at  Dl2:>  each  per  annum  ? 

191.  What  is  the  greatest  common  measure  of  82  &  124  ? 

192.  What  is  the  greatest  common  measure  of  !  64  &  248  ? 
19  j.  What  is  the  least  common  multiple  of  3,  4,  and  5  ? 
194.  (  f  ',  5,  6,  and  7? 

19  .  Of  2,  3,  5,  and  12? 

196.  Reduce  ifl  to  its  lowest  terms. 

197.  Reduce  i,  |,  and  |,  to  a  common  denominator. 
19^.  Reduce  12J  to  an  improper  fraction. 

199.  Reduce  y  to  a  mixed  number. 

200.  Reduce  f  of  |  of  f  to  a  single  fraction. 

201.  Reduce  |  of  a  penny  to  the  fraction  of  a  pound. 

202.  Reduce  f  of  a  Cwt.  to  the  fraction  of  a  lb. 

203.  Reduce  |  of  a  L.  to  its  value. 

204.  Reduce  "?Jd.  to  the  fraction  of  a  shilling. 

205.  Add  f  and  f . 

206.  Add  I  and  |. 

207.  Add  I  of  a  shilling,  and  j^j  of  a  penny. 

208.  From  |,  take  4. 

209.  Take^o  from'}. 

210.  Tell  the  product  off  by  f. 

211.  Off  by  f. 

212.  Divide  V  by  f . 

213.  Divide  f  by/_. 

214.  Tell  the  sum  of  29-0146+qi46-5+2109+-624l7-+ 
14-  6. 

21  >.  Tell  the  difference  between  91*73,  and  2-138. 

2 1 6.  Tell  the  product  of  79-347  by  23-15. 

217.  Of5i-3  by  1 000. 

218.  Tell  the  quotient  of  27  by  -2685. 

219.  Of  2  7-3  by  100, 

220.  Reduce  ^V  ^^  a  decimal. 

221.  Reduce  9d.  to  the  decimal  of  a  Z. 

22  \  Reduce  I  dwt  to  the  decimal  of  a  L. 
223  Reduce  15s.  9f  d.  to  the  decimal  of  a  L, 
224.  Tell  the  value  of  -625  of  a  shilling. 


Questions,  1 1)9 

225.  Of -8635  of  a  L. 

226.  Of  62j  of  a  Cwt. 

227.  If  4  yds  cost  1 2s.,  what  cost  8  yds.  9 

228.  if  7  yds.  cost  I  5s  ,  what  cost  9  yds.  ? 

22  ^  if  :;6  men  cm  build  a  wall  in  2i  days,  how  many  men 
can  do  it  in  '^  i  days  ? 

230.  if  6  Cwt.  i  qr.  of  sugar  cost  1^1 8  ..  1 6  ..  4,  what  cost 
3  Cwt.  I  qr.  27lb.  ? 

QM.  If  J  8  yds.  costnOs,,  what  cost  31  yds.  ? 

232.  If  50  men  can  perform  a  piece  of  work  in  12  days, 
how  many  can  do  it  in  4  days  ? 

^33.  What  will  12  yds.  of  lace  cost,  at  the  rate  of  Xr56  for 
96  yds.? 

231.  If,  when  the  price  of  wheat  is  ^s.  6d.  a  bushel,  the 
penny  loaf  weighs  8  oz.,  wiiat  must  it  weigh  when  the  price 
of  wheat  is    6s.  a  bushel  ? 

235.  How  many  men  must  be  employed  12  days,  toper- 
form  the  work  which  4  men  can  do  in  48  days  ? 

236.  If  l^  yds.  cost  9s.,  what  cost  2  I  yds.  ? 

237.  A  lent  B  T^O  dollars  for  8  months;  how  long  must 
B  lend  A  500  dollars   to  be  equivalent  ^ 

238.  How  many  yds.  can  be  bought  for  Ll4..  8,  when  l6 
yds., cost  1:  s.  ? 

.<39.  A  giddsmith  bought  14  lb.  3  oz.  8  dwt.  of  gold,  for 
2035  dollars,  what  is  that  per  ounce  ? 

240    If  a  staiT      feet  high,  casts  a  shade,  on  level  ground, 
5  f -et  long,  how  high  is  that  steeple,  the  shade  of  which,  at  the 
same  ti-me,  measures  -41  5  feet? 
k        2^1.   In  how  mny  days  can    12  men  perform  apiece  of 
1  work,  wliich  1 8^  men  can  do  in  '»0  days  ? 
I      242.   In  i8i 8,  the  number  in  the  Moravian  societies  was 
f  stated  to  be  ;  6000,  and  that  they  then  had  170  missionaries  & 
f  a!?sistants  among  the  heathen.     If  the  rest  of  the  nominally 
Christian    world   did   as  well  as  the  Moravians,    how   many 
'     missionaries  would  now  be  in  the  tield  ?     (See  No.  14.) 
r        24  >.   How   nia;»y  of  those  destitute  of  the  gospel,  would 
fall  to  tu'  lot  of  e.u'h  missionary?     (See  Nos.  13  and  14.) 

:^44  If  the  whole  number  of  Protestants  is  60  millions, 
and  they  had  all  done  as  well  as  the  Moravians,  how  many 
missionaries^  would  they  have  in  the  field? 

24  •.  How  many  would  fall  to  the  lot  of  each  missionary, 
in  that  case  ? 

246.  The   Moravians  reckoned   their  converts  from  hea- 
i  thenisni  to  be  60200  ;  if  the  rest  of  the  Protestaat  world  had 

P 


\ 


170  ({uestions. 

done  as  well,  what  would  have  been  the  whole  number  oi 
Gonverts  ? 

247.  The  Baptist  mission  at  Serampore,  was  begun  in 
1794;  and  in  1818,  they  had  60  native  preachers,  and  20 
churches  of  converted  natives  ;  of  which,  that  at  Chittagong 
was  estimated  at  150  members  ;  that  at  Jessore,  95  ;  that  at 
Dinagepore,  105  ;  that  at  Serampore  and  Calcutta,  190 :  how 
many  in  these  four  ? 

248.  If  the  other  16  contained  half  as  many  in  proportion, 
what  would  be  the  whole  number  ? 

249.  The  number  of  slaves  imported  into  Havana  from 
Africa,  from  Dec.  1,  1816,  to  July  Si,  1817,  was  i  1161 ;  how 
many  would  that  be  in  a  year,  at  the  same  rate  ? 

250.  If  4  men,  in  12  days,  can  reap  c  6  acres  of  wheat, 
how  many  acres  can  6  men  reap  in  18  days  ? 

2:>  .  If  6  men  can  reap  72  acres  in  12  days,  how  many 
men  can  reap  96  acres  m  h  days  ? 

252.  If  the  wages  of  6  men  for  42  weeks  be  60  dollars, 
what  will  be  the  wages  of  14  men  52  weeks  ? 

255,  If  40  shillings  be  the  hire  of  8  men  for  4  days,  how 
many  days  must  '60  men  work  for  ^20  ? 

254.  If  1 2  oxen  eat  24  acres  of  grass  in  1 5  days,  how  many 
acres  will  serve  24  oxen  45  days  ? 

255.  If  the  interest  of  350  dollars  for  8  months  is  18  dol- 
lars, what  sum  in  a  year  will  gain  6  dollars  ? 

2.6.  If  100  dollars  in  a  year  gain  6  dollars,  what  will  /i25 
dollars  gain  in  254  days  ? 

257.  If  LlOO  gain  Zr6  in  12  months,  what  sum  will  gain 
is  6  in  6  months  ? 

258.  An  auxiliary  missionary  society,  composed  of  100 
slaves,  at  Berbice,  S.  America,  raised,  in  8  months,  LS5  ster- 
ling ;  what  would  that  be  for  each  member  per  annum  ? 

259.  Iff  of  a  yd.  cost  |  of  a  £.,  what  cost  y^o  of  a  yd.  ? 

260.  If  j\  of  a  ship  is  worth  ^273  ..  2 ..  6,  what  is  -^^  of  it 
worth  ? 

261.  If  the  penny  loaf  weighs  6y\oz.  when  wheat  is  5s.  a 
bushel,  what  ought  it  to  weigh  when  wheat  is  8s.  6d.  a  bush.? 

262.  If  3  men;  in  6  days,  spend  ZilOf,  what  will  20  men 
spend  in  30  days  ? 

263.  if  1-5  oz.  of  silver  costs  7*8s.,  what  cost  29'1  lb.? 

264.  What  cost  S-4  lb.  at  ^£4-5  for  1-47  Cv/t.  ? 

265.  If  the  wages  of  i  men  for  y.4»6  days,  be  ^18-9,  what 
will  be  the  wages  of  8  mea  tor  16*4  days  ? 


Questions. 


171 


£66. 
267. 
268. 
269. 
270. 
271. 
272. 
273. 
274. 
2^5. 
276. 

289, 
290. 
29  i. 
292. 
293, 


Find  the  cost  of ' 
7612  lb.  at  id. 
6812 
4712 
15344 
7672 
9424 
8652 
1218 
8612 
7813 
1218 
6002 


Find  the  cost  of 

121  lb.  at  ^s, 


. 8  per  lb. 
per  lb* 


278. 
id.      279.         471  5s. 

|d.      280.         191  6s. 

Id.      281.         242  8s. 

id.      282.  345  6s.  8d. 

Id.      283.  678  13s.  4d. 

lid.      284.         567  13s.  2d. 

2id.      285.  825     ^3..    6 ..    8 

IJd.      286.  676       14..  17..    9i 

3id.      287.         346      12..  l9..  Hi 
8d.      288.  488       14..    8..    6 

llid. 
Find  the  cost  of  ^^7lb.  lOoz.  Troj,  at  ^1 ..  6 
Of  13  lb.  10  oz.  12  dwt.  8  gr.at  £2 ..  3  ..  4 
Of  476  A.  3  R.  28  p.  at  ^4 ..  12 ..  8  per  acre. 
Of  957  A.  3  R.  16  p.  at  ^3  ..  7  ..  1 1  per  acre. 
F.nd  the  neat  weight  of  856  Cwt.  1  qr.  19  lb.  of  to- 
bacco, tare  in  the  whole,  17  Cwt.  2  qr.  13  lb. 

294.  Find  the  neat  weight  of  1 3  hhds.  of  tobacco,  each 
weighing  6  Cwt.  2  qr.  27  lb.  gross  ;  tare  in  the  whole,  9  Cwt. 
S  qr.  17.  lb. 

295.  Find  the  neat  weight  of  6  casks  of  sugar,  weighing, 
gross,  4  C\vt.  3  qr.  12  Ib/cach  ;  tare  \B  lb.  each. 

296.  What  do  they  come  to,  at  £9,  .,4. .7  per  Cwt.  ? 

297.  Find  the  neat  weight  of  12  casks  of  raisins,  each 
weighiRg  S  Cwt.  3  qr.  26  lb-  gross  ;  tare>  20  lb.  per  cask. 

298    What  is  the  value,  at^:2  ..  14  ..  6  per  Cv/t.  ? 

299.  Find  the  neat  weiglit  of  50  kegs  of  rigs,  gross  86  Cwt. 
k  6  qr.  14  lb.,  tare  14  ib.  per  Cwt- 
\      300,  What  do- they  ccuie  to,  at  I9s.  8d.  per  Cwt.  ? 

301.  In  28  bags  of  coifee,  each  -  Cwt.  3  or.  12  lb.  gross, 
e  14  lb.  per  Cwt-,  trett  4  lb.  per  104  lb-,  how  much  aeat  ? 

302.  Find  the  value,  at  dg%  ..  18  ..  9  per  Cv^t. 

303    In  9  Cwt.  3  qr.  27  U).  gross,   tare  38  ib,   trett  4  lb. 
\  per  104  lb.,  how  many  lb.  neat  .' 
1       304-.  Find  the  value  at  8id   [)er  lb. 

30 "5.  Tell  the  interest  of  Li 57  for  1  year,  at  6  per  cent 

30*-:.    (if  7^25  i,  at  4  percent. 

3^7-  Of 'L5678,  at  7  per  cent  ? 

308.  Of  ^234,  at  7-J  per  cent. 

309.  Of  S 158,  at  S  per  cent. 
3 1  J.  Of  S789,  at  5}y  per  cent 

V      31 1.  Of  Z/2340,  for~i4Tionths,  at  6i  per  cent 


72  Questions- 

S12.  Of  L600  for  8  months,  at  l^f  per  cent. 

313.  Of  Z6740  for  7  montlis,  at  ej  per  cent. 
n4.  Of  jL56  ..  12 ..  8  for  I J  years,  at  5  per  cent 
il5.  Of  L65  ..  19  ..  6  for  5  years,  at  6  per  cent. 

SI 6.  Of  L66  ..  10  ..  6  for  6  years,  at  6^  per  cent. 

317.  Tell  the  amount  of  8624-25,  for  130  days,  at  6  per  ct. 

SIS.  iff  8786-30,  for  235  days,  at  6i  per  cent. 

319.  Of  g687-34,  for  320  days,  at?  per  cent 

S20.  Of  2.62S  ..  13  ..  S3^  for  5  y.  IH  nio-  at  6  per  cent 

32 1.  What  is  the  insurance  of  an  East  India  ship  and 
<^rgo,  valued  at  L8406  ..  1 8  ..  6,  at    5^  per  cent. 

322.  What  principal,  at  interest  for  8  years  at  5  per  cent, 
will  amount  to  i>720  ? 

323.  At  what  rate  per  cent  will  Z600  amount  to  jL9ii4,in 
9  years  ? 

324.  In  what  time  will  X700  amount  to  L940,  at  5  per  ct.  ? 
325*  A  merchant  has  sold  goods  on  commission,   to  the 

amount  of  S3 45 600  ;  what  is  his  commission,  at  2|  per  ct.  ? 

326.  What  sum  at  interest  for  9  years  and  6  months,  at  7 
percent,  wdl  amount  to  Sl456'87i  v 

327.  If  2i  per  cent  is  allowed  for  commission,  what  must 
be  paid  on  1/1234..  17..  8? 

328.  In  1817,  t^ngland  agreed  to  pay  Spain  X400000  ster- 
ling, for  which  Spain  agreed  to  abolish  the  slave  trade,  after 
May,  i  820 ;  what  is  the  annual  interest  of  that  sum,  at  6  per 
cent  ? 

329.  In  1818,  the  Connecticut  school  fund  was  §1608673 
•39  ;  what  would  it  yield  annually  at  6  per  cent? 

330-  The  amount  distributed  from  the  common  school 
fund  of  New-York,  was  §1^0000  ;  what  must  the  fund  have 
been,  to  yield  that  amount  at  7  per  cent? 

331.  The  expenditures  on  the  United  States  armory  at 
Springfield,  from  179.0  to  I8;i0,  had  been  82072676;  if  this 
had  been  put  into  a  fund  for  religious  purposes,  what  would 
it  annually  yield  at  6  per  cent  ! 

532.  How  many  bibles  would  it  annually  furnish  for  cha- 
ritable distribution,  at  60  cesits  each  ? 

333.  How  many  young  men  would  it  assist  in  obtaining  an 
education,  at  gl25  each  ? 

334.  How  many  missionaries  would  it  support,  at  S500 
each  per  annum  ? 

33^.  In  18:^0,  the  permanent  fund  of  the  Connecticut 
Missionary  Soc-  was  D33405-55J- ;  what  sum  will  it  annually 
yield,  at  6  per  cent  ? 


^iiesiionB'  173 


m 

»      3S^.  How  many  weeks  missionary  labor  will  it  pay  for 
'    ever/  year,  at  8  dollars  a  week  '' 

S  r  .  In  1 820,  the  amount  ol  interest  received  by  the  Ame- 
rican Board  for  Foreign  Missions,  was  D2 154-60  ;  what 
must  have  been  the  principal  of  their  permaneat  fund,  to  yield 
that  amount,  at  6  per  cent  ? 

338.  What  is  the  amount  of  L720,  for  4  years,  at  5  per 
cent,  compound  interest? 

339.  What  is  the  amount  of  L50,  in  .'i  years,  at  5  per  cent, 
compound  interest? 

340.  What  is  the  compound  interest  of  1/370,  for  6  years, 
at  4  per  cent  ? 

341.  What  is  the  compound  interest  of  i/450,  for  7  years, 
at  5  per  cent  ? 

342.  Tell  the  present  worth  of  L80..  15,  for  19  months, 
discount  at  5  per  cent. 

343.  Tell  the  discount  of  jL159I  ..  2  ..  4,  for  1 1  months,  at 
6  per  cent. 

o  14.  Sold  goods  for  LS97  ..  15  ..  7,  to  be  paid  in  4  months  ; 
what  is  the  present  worth,  at  3|  per  cent  ? 

345.  A  owes  B  -L2408,  of  which  L1234  is  payable  in  6 
months,  and  the  rest  in  1 0  months ;  but  they  agree  to  reduce 
them  to  one  payment ;  when  must  that  be  ? 

346.  A  merchant  has  owing  to  him  LIOOO,  of  v/hich  Irl50 
is  due  now,  LI 50  in  2  months,  L200  in  4  months,  and  the  rest 
in  6  months  ;  what  is  the  equated  time  ? 

347.  C  owes  D  7.480,  payable  5  months  hence ;  but  is 
willing  to  pay  L80  now,  if  D  will  wait  longer  for  the  rest ; 
to  what  time  must  he  vvait  ? 

S'iS.  What  quantity  of  tea,  at  20s.  per  lb.  must  be  gives, 
in  barter  for  1  Cwt-  of  chocolate,  at  4s.  per  lb  ? 

3  v9.  How  much  flour,  at  5t)s.  per  Cwt-  must  be  given  for 
7  Cwt-  of  raisins,  at  5d.  per  lb-  ? 

350.  A  has  24  sheep,  at  1 6s.  8d.  each,  for  v»'hich  B  is  to 
pay  L12  in  cash,  and  the  rest  in  potatoes,  at  2s.  a  bushel  ; 
how  many  bushels  of  potatoes  must  A  receive  ? 

351.  B  delivered  6  hhds.  of  molasses,  at  6s.  8d.  a  gallon, 
to  C,  for  2.S2  yds.  of  cloth  ;  what  was  the  cioth  per  yard  ? 

352.  How  much  coflee,  at  25  cents  per  lb.  can  I  have  for 
56  lb.  of  tea,  at  80  cents  per  lb.  ? 

^^5S.  A  delivered  to  B  9Sii  bushels  of  corn,  at  50  cents  a 
bushel,  and  received  55  Cwt.  2  qr.  of  cheese,  at  4  dollars  per 
Cwt. ;  how  much  money  must  A  receive  in  addition,  to  pa 
for  his  corn  ?  P2 


IT4  Questions' 

364.  How  much  wine,  at  Dl'^28  per  gal.  must  I  liave  for  13 
Cwt.  1  qr.  7  lb.  of  raisins,  at  D9'444  per  Cwt.  ? 

355.  If  I  buy  candies  at  '9  cents  a  lb.,  and  sell  them  at 
23  cents  a  lb.,  what  shall  1  gain  per  cent  ? 

356.  Bought  indigo  at  DM0  a  lb.,  and  sold  it  at  90  cents 
a  lb. ;  what  was  lost  per  cent  ? 

557.  Bought  74  gals,  of  wine,  at  DM0  a  gal.,  and  sold  it 
for  DcSO;  what  was  gained  or  lost  per  cent? 

558.  Bought  hats  at  8s.  each,  and  sold  them  at  9s.  6d.  each ; 
what  was  gained  per  cent  ? 

559.  If  1  buy  wheat  at  Dl-25  per  bushel,  how  must  I  sell 
it,  to  gain  18  per  cent  ? 

560.  If  a  hhd.  of  rum  cost  50  dolls,  for  how  much  must  it 
be  sold,  to  lose  iO  per  cent  ? 

361.  If  60  lb.  of  steel  cost^^JLS  ..  10,  how  must  I  sell  it  per 
lb.  to  gain  15  J  per  cent  ? 

862.  A  and  B  join  stock,  and  make  up  D600.  A  yiuts  in 
B225,  and  B  the  rest.  They  gain  D150  ;  what  is  the  share 
of  each  ?  ^ 

563.  A  man  dying,  left  5  sons,  as  follows  :  A,  184  dolls., 
B  155,  and  C  96  ;  but  when  his  debts  were  paid,  there  were 
but  368  dolls,  left;  what  is  the  share  of  each  r 

304.  A  and  B  companied  ;  A  put  in  irl35,  and  took  |  of 
the  gain  ;  what  did  B  put  in  ? 

365.  A,  B,  and  C,  entered  into  partnership.  A  put  in 
D170  for  8  months,  B  Dl20  for  10  months,  and  C  1)2  lO  for 
5  months;  and  they  lostD82;  what  is  each  man's  share  of 
the  loss  ? 

366.  A,  B,  and  C,  hold  a  pasture  in  common,  for  which 
they  pay  L40  a  year.  In  this  pasture,  A  had  80  oxen  76 
days,  B  7S  oxen  50  days,  and  C  100  oxen  90  days ;  what 
must  each  man  pay  ? 

567.  in  I8l7,  the  London  Hibernian  Society  had  in  its 
schools  in  Ireland,  under  gratuitous  instruction,  52000  poor 
children,  at  an  average  expense  to  the  Society  of  5s.  sterling 
each  ;  what  is  the  amount  in  Federal  money  ? 

368.  The  return  oi  Bonaparte  to  France  from  Elba,  and 
his  subsequent  measures  to  the  battle  of  Waterloo,  are  stated 
to  have  cost  the  French  nation  i02l  millions  of  francs  ;  how 
much  is  that  in  Federal  money  ? 

569.  In  18  i  8,  the  London  African  Institution  for  promo- 
ting the  abolition  of  tl^e  slave  trade,  expended  LS05  ..  15  ..  9 
vsterling ;  how  much  is  that  in  New- York  currency  ? 


Questions.  175 

H^fpO.  The  bank  of  England  is  said  to  have  a  capital    of 
^^T!m60O500  sterling  ;  how    much   is   that  in    ^iew- England 
currency  ? 

371.  In  18)9,  the  London  Prayer  Book  and  Homily  Soc. 
expended  i.!iC06  ..  11  ..  4  sterling ;  how  much  is  that  in  Penn- 
sylvania currency  ? 

oTz,  The  London  Hibernian  Soc.  expended  7  8387  ..  16  ..  8 
sterling;  how  much  is  tiiat  in  fcouth-Carolina  cunency  ? 

373.  The  receipts  of  the  Biitish  Naval  and  Military  Bible 
Soc.  were  L'^'i69.  sterling  ;  how  much  is  that  in  Canada  cur.? 

374.  in  I8l9,  the  exports  of  Russia  were  to  the  amount  of 
43559343  rubles  more  than  their  import:?,  which  were  167 
millions  of  rubles;  what  is  the  amount,  in  Federal  money, 
of  their  exports? 

375.  The  expenditures  of  the  Scottish  Miss.  Soc.  for  1819, 
were  L4599  ..  11  ..  1 1  sterling  ;  what  is  that  in  >Jew-Jersey 
currency  ? 

37^^.  The  receipts  of  the.  New  Hampshire  Miss.  Soc.  for 
1820,  were  D^537-21  ;  what  is  that  in  sterling/ 

377.  The  receipts  of  the  Boston  Jews  Soc.  were  D 1 195*67  ; 
liQ^w  much  is  that  in  Virginia  currency  ? 

378.  The  expenditures  of  the  British  and  Foreign  School 
Soc.  were  jL2432  ..  3  ..  3  sterling;  hov/  much  is  that  in  North 
Carolina  currency  ? 

379.  The  Berbice  Auxiliary  Miss.  Soc.  composed  of  slaves, 
contributed,  in  1820,  420  guilders;  how  much  is  that  in 
Federal  money  ? 

380.  The  total  expenditure  of  the  British  and  Foreign 
Bible  Society  in  17  years,  was  jL908248  ..  10..  6  sterling  ; 
how  many  livres  is  that  ? 

381.  How  much  in  Federal  money  ? 

382.  The  receipts  of  the  Maine  Miss  Soc.  in  1820,  were 
D20.58-47  ;  how  much  is  that  in  Georgia  currency  ? 

383.  The  receipts  of  the  Hampshire  Missionary  Soc  were 
D I  590-59  ;  how  much  is  that  in  Canada  currency? 

384.  The  receipts  of  the  New- York  Miss.  Soc  in  1807, 
were  D  1.360*47  ;  how  much  is  that  in  Irish  currency  ? 

335.  The  receipts  of  the  Vermont  Missionary  Soc  in  1812, 
were  D652-67  ;  how  many  livres  is  that  ? 

386.  The  receipts  of  the  Connecticut  Miss.  Soc-  in  1816, 
w^ere  D6019-32;  how  many  rubles  is  that  ? 

387.  The  receipts  of  the  Connecticut  Bible  Soe.  iu  1816^ 
w^ere  D2 1 77*20  ;  how  many  rials  of  plate  is  that? 


176  ^estions. 

3S8.  The  receipts  of  the  Massachusetts  Miss.  Socin  181S, 
were  D3120'04  ;  how  much  is  that  in  sterling  '' 

o89.  The  receipts  of  the  Massachusetts  Society  for  pro- 
pagating the  Gospel,  in  1810,  were  D2477'8G;  how  much  is 
that  in  New-England  currency? 

390.  The  expenditures  of  the  Massachusetts  Society  for 
promoting  Christian  Knowledge,  in  1813,  were  Di 407*62; 
how  much  is  that  in  Delaware  currency  ? 

391.  The  receipts  of  the  Berkshire  and  Columbia  Miss. 
Soc.  in  1 813,  were  Dsi  1*44  ;  how  much  is  that  in  New -York 
currency  ? 

392.  The  receipts  of  the  American  Board  for  Foreign  Mis- 
sions the  first  year,  were  Dl3»9*52 ;  how  much  is  that  in 
South-Carolina  currency  ? 

393.  The  second  year,  D139j304;  how  much  is  that  in 
Irish  currency  ? 

S94.  The  third  year,  Dl  1436*  18;  how  much  is  that  in 
Canada  currency  ? 

395.  How  much,  in  Federal  money,  did  David  give  Arau- 
nah  for  his  threshing  floor  and  oxen,  (2  Sam.  24-  24,)  it  being 
50  shekels  of  silver  ? 

396.  How  much,  for  the  whole  place,  (1  Chron.  21.  25,)  it 
feeing  60(J  shekels  of  gold  ? 

397.  What  was  the  avoirdupois  weight  of  Absalom's  hair, 
(2  Sam   14.  26,)  it  being  200  shekels  ? 

398.  How  many  bushels  of  flour,  and  how  many  of  meal, 
were  required  daily  for  Solomon's  table,  (l  Kings  4.  2S,)  it 
being  30  cors  of  flour,  and  ^^O  of  meal  ? 

399.  What  were  the  dimensioni<,  in  feet,  of  Solomon's 
temple,  it  being  60  cubits  long,  20  wide,  and  30  high  ? 

400  How  many  wine  gallons  did  the  brazen  sea  contain, 
being  3000  baths  ? 

4  1.  How  many  miles  was  Bethany  from  Jerusalem,  being 
15  Hebrew  furlongs  ? 

402.  How  many  acres  of  land  were  assigned  to  the  Le- 
^ites  as  glebes,  (Lev.  35.  3 — 5,)  being  lOOo  cubits  square  on 
each  of  the  4  sides  of  each  of  the  48  Levitical  cities  ? 

403.  George  W^ashington  was  born  Feb-  22,  1732,  and 
died  Dec.  14,  1799  ;  how  old  was  he  ? 

40',.  The  p(»pulation  of  the  New -England  states,  at  each 
census,  was  as  iEbllows : 


(luesiions. 


irr 


Vermont, 

N.  Hampshire, 

Miiiue, 

Massachusetts, 

Rhode  Island, 

Connecticut, 


New  York, 

New-Jersey, 

Peunsjlvariia, 

Delaware, 

Maryland, 

Dist.  Columbia, 


In  1810. 

In  t8'20. 

2 '7895 

23; 7 04 

2    ^460 

21-4 -61 

228705 

29^-^335 

47  040 

52^-287 

76931 

83059 

261942 

275248 

In  1790.  In  1800. 

85589  154449 

141885  183858 

9r>540         151719 
3787  7         422530 
68825  69122 

237946  251002 

What  was  the  number  of  inhabitants  in  New  England  at 
each  census  ? 

405.  The  population  of  the  Middle  States  at  each  census, 
was  as  ft)llows  : 

340 1 20  586050 

184139  2    H49 

434  373  602545 

69094  64273 

319728         3i96&2 
8124 
AVhat  was  the  number  of  inhabitants  in  the  Middle  States, 
at  each  census  r 

406.  The  population  of  the  Southern  States  at  each  census, 
was  as  follows : 

886149 
478(03 
S45591 
162686 


959049 

1372812 

245562 

277575 

810091 

1049P^8 

72674 

72749 

380546 

407350 

2402  5 

33039 

747610 

393751 

239073 

82548 


974622 
555500 
415115 
252433 


lO'^S.^e 


Virginia, 
K.  Carolina, 
S.  Carolina, 
Georgia, 
Alabama, 

What   was  the  number  of  inhabitants  in  the 
States,  at  each  census  ? 

407.  The  population  of  the  Western  States  at  each  census, 
was  as  follows : 


^38829 
502741 
340989 
127901 

Southern 


Kentucky, 

73677 

22095         4065 119 

664317 

Tennessee, 

35691 

14260         261727 

4:22813 

Ohio, 

42179          230760 

581434 

Mississippi, 

40352 

75448 

Louir^iana, 

76556 

153407 

Indiana, 

4875           24520 

147178 

Illinois, 

12282 

55211 

Missouri, 

20845 

66586 

Michigan  Ter. 

4762 

8896 

Arkansaw  Ter 

14273 

What  was 

the  number  of  inhabitants  in  the 

Western 

States,  at  each 

census  ? 

408.  What 

was  the  whole  number  of  inhabitants  in  the 

Wnited  States, 

at  each  census  ^ 

178  (luestiotis, 

409.  The  contributions  to  the  American  Board  for  Foreign 
Missions,  for  the  year  ending  Aug.  31,  1820,  were,  from  the 
several  states,  as  follows  :  Massachusetts,  $1466l'7l  ;  Con- 
necticut, $6036-68  ;  New  York,  $379i-41 ;  Vermont,  $1843 
•69;  New-Hampshire,  S1464.-82;  Maine,  |  451-83;  New- 
Jersey,  $1443-51  ;  Georgia,  81280*52  ;  S.  Carolina,  $764-30  ; 
Pennsylvania,  $702-30  ;  Maryland,  $68-  -50  ;  N.  Carolina, 
$r}64-3£;  Ohio,  $392  91;  Tennessee,  $251-17;  Virginia, 
$209  ;  Louisiana,  $200  ;  Rhode-Island,  $.11-56;  Delaware, 
Si  05-44;  Mississippi,  $£0  ;  Dist.  of  Columbia,  $10;  Choc- 
taw Nation,  Sl69;  Cherokee  Nation,  $8  ;  places  unknown, 
$4   1-97.     How  much  in  all  ? 

4  lO.  How  much  less  than  half  the  whole,  was  contributed 
by  Massachusetts  ? 

4H.  How  much  more  than  half  the  whole,  by  Massachu- 
setts and  Connecticut  ? 

4 1  *.  If  the  amount  contributed  by  Massachusetts,  were 
divided  equally  among  the  inhabitants  of  that  state,  accor- 
ding to  the  census  of  » 8  20,  (See  No.  404,)  how  much  would 
it  be  for  each  ? 

413.  If  the  amount  contributed  by  Connecticut,  were  so 
divided  among  the  inhabitants  of  that  state,  what  would  it  be 
for  each  ? 

41 4.  If  New- York  had  contributed  in  the  same  proportion 
as  Massachusetts,  how  much  would  have  becH  raised  in  that 
state  ' 

415.  If  all  the  states  had  contributed  in  the  same  propor- 
tion as  Massachusetts,  what  suui  would  have  been  raised  iu 
the  whole  ? 

416.  If  the  sum  which  was  contributed,  enabled  the  Board 
to  support  88  missionaries  and  assistants  among  the  heathen, 
what  is  that  for  each  ? 

417.  If  all  the  states  had  contributed  in  the  same  propor- 
tion as  Massachusetts,  how  many  would  it  have  enabled  the 
Board  to  support,  at  that  rate  ? 

418.  In  1810,  the  number  of  blacks  in  the  United  States, 
was  as  follows:  Northern  states,  3 1 687  slaves  91317  free; 
Southern  spates,  1159677  slaves,  95129  free:  how  many 
more  slaves  than  free  ? 

419.  In  181'^,  there  were  living  1336  ministers  of  the  gos- 
pel, graduated  at  the  following  colleges,  to  wit:  Harvard, 
Yale,  C9lum-)ia,  Brown»  Dartmou^^h,  Carlisle,  Williams, 
UBion,  Bowdein,  Middlebury,  South-Carolina,  Transylvania, 


Questions.  179 

William  &  Mary  ;  allow  130  more  for  Princeton,  and  84  for 
other  colleges  in  America  and  abroad  ;  and  how  many  mi- 
nisters are  there  in  the  United  States,  who  have  been  educa- 
ted at  college  ? 

420.  If  the  number  of  ministers  who  have  obtained  a  suffi- 
cient education  without  going  to  college,  is  half  as  many  ; 

,how  many  educated  ministers  are  there  in  the  U.  Stales  ? 

421.  If  900  of  these  are  in  New- England,  how  many  more 
are  wanted  there,  to  make  one  for  every  800  souls  ?  (See 
No.  4«  4. 

^22.  How  many  are  wanted  in  the  other  states,  at  the 
same  rate  ?     (See  No.  -t08.) 

4:1^.  How  many  souls  are  there  in  New-England,  to  one 
educated  minister  ? 

424.  How  many  in  the  other  slates  ? 

425.  The  population  of  the  United  States  is  said  to  have 
doubled  once  in  'iS  years  ;  if  it  should  continue  to  do  so, 
what  wdl  it  be  in  the  year    958  ? 

4-6.  The  number  of  ministers  educated  at  college  in  the 
United  States,  has  doubled  once  in  TO  years  ;  suppose  t  to 
continue  to  double  once  in  69  years,  and  the  numbei  of  edu- 
cated ministers  in  1820  to  be  25l'0 ;  what  will  it  be  in  the 
year  1958? 

427.  If  one  minister  is  necessary  for  every  800  souls,  how 
many  will  then  be  destitute  ? 

4.  8.  How  many  will  be  destitute,  for  one  that  is  supplied? 

429.  Mrs.  Norris  left  §30000  to  the  American  Board  for 
Foreign  Missions  ;  what  is  the  annual  interest,  at  6  per  cent? 

430.  How  many  missionaries  will  that  support  continually, 
at  $500  each  per  annum  ? 

431.  If  each  of  these  missionaries  should  be  instrumental 
in  the  conversion  of  20  heathen  souls  every  year,  what  will 
be  the  whole  number  in  a  century  ' 

4S2,  If  that  interest  should  be  applied  to  the  board  and 
education  of  heathen  children  in  the  mission  families  at  Cey- 
lon, at  12  dolls,  each,  how  many  would  it  constantly  support? 

433.  How  many  heathen  children  would  thus  receive  a 
Christian  education,  in  a  century,  allowing  each  to  be  4 
years  at  school  ? 

434.  If  it  should  be  applied  to  the  school  expenses  of  such 
children  as  are  fed  and  clothed  by  their  parents,  which  at 
Ceylon  amount  to  48  cents  each  per  annum,  how  many  such 
children  would  it  keep  at  school  continually  ? 


180  Questions. 

435.  How  many  heathen  children  would  be  educated  bv 
it,  in  this  way^  in  a  century,  allowing  each  to  be  4  years  at 
school r         - 

,S5.  The  exhibition  of  West's  picture  of  Christ  healing 
the  sic^,  produced  the  sum  of  '|4l33'4o  to  the  Pennsylvania 
hospital,  in  ;8i8,  froui  165  5  visitors;  how  much  is  that 
from  each,  in  Pennsylvania  currency  ? 

437.  In  the  year  i274,  the  price  of  a  small  bible,  neatly 
written,  was  ^35  sterling;  and  the  wages  of  a  laboring  man 
were  i  Jd.  per  day  :  huw  many  years,  of  313  working  days 
Cach,  must  h:*,  have  labored  to  pa^  for  a  bible  ? 

4  >8.  If  a  laboring  man  now  earns  50  cents  a  day,  how 
Kiany  bibles  woul:;  t  le  same  labor  pay  for.  at  bO  cents  each  ? 
4.^  .  If  oiie  person  in  >()  in  the  United  States,  wears  watch 
tiTiikets  that  cost  5  doll*  s,  how  much  might  be  saved  for 
doing  good  by  that  class  (f  ^eutlv-men,  by  doing  without  these 
useless  articles  / 

44.0.  If  one  in  200  should  save  10  dolls,  in  this  way,  what 
would  be  the  anKmnt  . 

'4  .  If  ')ne  in  1  0  should  save  ^^0  dollars  in  this  way, 
"what  woii'l  be  tne  arie.)iiii   ? 

44  .  If  one  in  (:•  0  should  save  50  d.rllars  in  this  way, 
whai  wo*. Id  'Ml  the  am'»unt  r 

4  3.  If  one  in  ^3!j'^0ii  should  save  100  duliars  in  tliih  way, 
what  would  be  the  amount  ? 

4  44.  What  is  the  whole  amount  ^hat  might  be  saved  bj 
these  fiV5  classes  of  gentlem  vn  ' 

4.5  If  clas-es  of  L  dies,  ecjiudly  rjumerous,  v/ear  orna- 
ments to  half  that  a*n  :«;.ni,  \vh   r  :\n^'  ih'.'V  ^^ave  ? 

446  A<id  together  the  -iv  ng  of  the  lariies  and  gentlemen, 
as  above  stated,  and  tell  what  is  the  animal  interest  it  would 
yield,  at  6  per  cent  ? 

4.i7.   How  many  young  men  would  it  assist  in  ob^ai 'n>!g 
an  educa^tum  for  the  gospel  oiini^iry,  at  A  ^25  each  per  an   ? 
4  8.   How  many  heathen  children  niight  be  ed'icated  by  It 
in  a  ceiury,  at  the  rate  mentioned  in  N  s.  i.s  l.  and  4  j 

4 19.  How  many  billies  would  it  annually  aiford  for  cha- 
ritable distrib  I  tion,  at  60  cents  each  ? 

4  50.  The  first  society  in  P — ,  N.  Y.  the.  population  of 
whi^  h  does  not  exceed  450  souls,  besides  supporung  the  g^js- 
pel  a-  home,  contributed  for  the  spread  of  the  j^o-pel  ahiui^d, 
in  the  year  1821,  the  fdlowing  sums,  to  wit:  For  ioni-stic 
Bftissious,  DiS-i^o;  cash  to  the  American  Board,  Dibo-dT; 


Questions,  ISi 

clothing,  5cc.  for  Indian  missions,  D341'35  ;  Auburn  Semi- 
ndry,  iJ39 ;  EJu':ation  a.id  Bible  societies,  estiuiated  at  D25  ; 
board  and  tuit.oii  ot  a  charity  student,  Do8  ;  clotUmg  for  do. 
estimated  at  030  :  how  jiiuch  in  all  ? 

451.  it'  all  the  people  m  the  United  States  should  contri- 
bute in  tha  same  proportion  every  year,  what  would  be  the 
amount,  and  how  many  missionaries  would  it  support,  at  500 
dollars  each  ? 

452.  In  Oct.  1816,  the  money  divided  to  school  societies 
in  Connecticut,  from  the  school  fund,  was  D^OoOS'TS,  and 
in  March,  IBIT,  the  sa^ne  sum  ;  and  in  trie  same  time,  there 
was  paid  to  citizens,  as  colli^tors^  fees,  Dlo395-82;  in  t^e 
same  time,  the  amount  of  the  state  tax  payable  into  the  trea- 
sury, was  048645*81  ;  how  much  did  the  people  of  Con- 
necticut receive  from  the  treasury  that  year,  more  than  they 
paid  in  ? 

453.  A  druggist  mixes  togetlier  several  simples,  as  follows: 
First,  2  oz.  3  dr.  1  sc;  second,  3  oz.  ^  dr.  1 S  gr. ;  third,  4 
oz.  7  dr.  2  sc. ;  fourth,  1 1  oz.  6  dr.  19  gr. :  what  is  the  weight 
of  the  whole  coiiiposition  ? 

454.  A,  B,  and  C,  each  owe  me  230  dolls. ;  D,  E,  and  F, 
each  twice  that  sum  ;  what  is  the  amount  ? 

455.  Borrowed  of  A  £'25  ..  I6  ..  6^,  of  B  £27 ..  16  ..  8,  of 
0^54..  6..  ri;  and  paid  A  sgl4..  19.,  10,  B  =glO..  19..  9, 
C  ^48  ..  1 0  ..  1 1  ;  how  much  do  1  still  owe  ? 

456.  What  cost  1945  bbls.  flour,  at  D6'25  per  bbl.? 

457.  What  cost^l3  lbs.  at  4s.  6d.  per  lb.  ? 

458.  if  a  man  who  fails  in  trade,  owes  3765  dollars,  and  is 
able  to  pay  45  cents  on  the  dollar,  what  sum  will  his  creditors 
lose  ? 

459.  In  36  weeks  and  4  days,  how  many  seconds  ^ 

460.  Bought  3  horses  for  £l6  ..  17  ..  7  each,,  and  2  cows 
for  £5  ..  14  ..  7  each,  and  3  bushels  of  wheat  for  IBs.  lO^d. ; 
what  is  the  whole  amount  ? 

461.  Divide  jLll..  11..  3  by  3. 

462.  If  a  debtor  pays  7s.  6d.  on  the  pound  on  L5S78,  how 
much  will  his  creditor  receive  ? 

463.  If  6  men  have  LS  ..  10  for  4  days  work,  how  much 
inu^t  36  men  receive  for  18  days  work  ? 

464.  Tell  the  amount  ot  D507'25  in  3  mo.  at  7^  per  cent? 

465.  In  1764  nails,  how  many  ells  Flemish  ? 

466.  Divide  L7  ..  1 ..  9  by  :^7* 

467.  What  cost  7121  lbs.  at  15d.  per  lb.  ? 


182  Questions. 

468.  What  cost  2345  acres,  at  L2  ..  3  *.  6  per  acre  r 

469.  Bought  2 i  bales  of  cloth,  in  each  bale  l3  pieces,  and 
in  each  piece  25  ells  English,  4  qrs.  3  na. ;  how  much  in  all  ? 

470.  Find  the  greatest  common  measure  of  246  and  372. 

471 .  Reduce  m  to  its  lowest  terms. 

472.  Reduce  |,  |,  and  J,  to  a  common  denominator. 

473.  Reduce  14/^;^  to  aa  improper  fraction. 

474.  Reduce  *||2  to  its  equivalent  number. 

475.  Reduce  |  of  f  of  |f ,  to  a  single  fraction, 

476.  Reduce  f  of  a  farthing  to  the  fraction  of  a  £* 

477.  Reduce  f  of  an  acre  to  its  value. 

478.  Add  1,71,  and  i  off. 

479.  From  y\,"take  ^\. 

480.  Multiply  i,  |,  and  3,  continually  together. 

481.  Divide  ^f  byf. 

48^.  If  i  of  a  ship  is  worth  L7S  ..  1  ..  3,  what  part  of  it 
worth  XS.iO^  ? 

483.  If  a  man  performs  a  certain  journey  in  35|  days,  tra- 
velling 13f  hours  a  day  ;  how  many  days  would  he  be,  tra- 
velling 1 1  j%  hours  a  day  ? 

484.  What  quantity  of  cloth,  f  yd.  wide,  will  line  9^  yds. 
that  is  2i  yd.  wide  ? 

485.  If  L2  ..5..}  ..  2|  be  the  interest  of  L9.5  for  |  of  a 
year,  in  what  time  will  lAS^  gain  -LItV  ^ 

486.  Tell  the  difference  between  1-9  r85  and  2*73. 

487.  Tell  the  productof  3  by  •3. 

488.  Tell  the  quotient  of -48624097  by  179. 

489.  Reduce  |  to  a  decimal. 

490.*  Reduce  5  oz.  12  dwt.  I6gr.  to  the  decimal  of  a  lb. 

491.  Tell  the  value  of  •u0994o  of  a  mile. 

492.  What  cost  6»25  hhds.  of  wine,  at  l-2s.  a  pint  ? 

493.  If  2  persons  receive  4-625s.  for  one  day's  labor,  how 
much  should  4  persons  receive  for  5*25  davs'  labor  ? 

49  i.  Tell  the  product  of  4  ft.  7  in.  by  9  ft.  6  in. 

495.  Of  12  ft.  5  in.  by  6  ft.  8  in. 

496.  Of  35  ft.  4|r  in.  by  1 2  ft.  3  in. 

497.  Find  the  content  of  a  load  of  wood  6  ft.  4  in.  wide,  4 
ft.  high,  and  7  ft.  8  in-  long. 

498.  How  many  solid  feet  in  a  bale  of  cotton  7  ft.  6  in. 
loiflig,  3  ft.  B  in.  wide,  and  3  ft.  3  in.  thick  ? 

499.  Find  the  solid  content  of  a  stick  of  squared  timber, 
P,0  ft.  3  m.  long,  1  ft.  2  in.  broad,  and  1 U  in.  thick. 

500.  Find  the  cube  of  29. 


(Questions.  183 

)1.  The  square  of  624. 

502.  The  cube  of  101. 

503.  The  square  of  4*  16. 

504.  The  cube  of  3-5. 

505.  The  square  of  |. 

506.  The  cube  of  |. 

507.  The  4th  power  off.        , 

508.  The  square  root  of  29506624. 

509.  The  cube  root  of  48228-044. 

510.  The  4th  root  of  194481. 

511.  The  square  root  of  IJ. 

512.  The  cube  root  of  39304. 

513.  The  cube  root  of  4^|. 

514.  The  square  root  of  f-^\. 

515.  Bought  13  yds,  of  cloth,  at  2(1.  for  the  first  ydi,  4d. 
for  the  second,  6d.  for  the  third,  and  so  on ;  what  was  the 
amount? 

5  i6.  Bought  100  acres  of  land,  at  1  doll,  for  the  first  acre^ 
2  dolls,  for  the  second,  3  dolls,  for  the  third,  and  so  on  ;  what 
^vvas  the  amount? 

5  i  7.  Fourteen  persons  bestowed  charity  upon  a  beggar,  the 
first  giving  3d.,  the  second  6d.,  the  third  9d.,  and  so  on  ; 
what  did  thelast  person  give,  and  what  did  the  beggar  receiver 

518.  A  debt  is  to  be  discharged  at  sixteen  several  pay- 
ments, the  first  to  be  t/6,  and  the  last  to  be  L200  ;  what  is 
the  common  diiferencc  of  the  payments,  and  what  is  the 
whole  debt  ? 

519.  Sold  10  yds.  of  clolh,  the  first  for  3s.,  and  the  last  for 
L2  ..  8  ;  what  was  the  common  difierence  ? 

520.  A  certain  person  married  off  his  daughter  on  New- 
Year's  day,  and  gave  her  5  dollars  towards  her  portion,  pro- 
mising to  double  it  on  the  first  day  of  every  month  through 
the  year;  what  is  the  amount  ? 

521.  Bought  a  horse,  with  4  shoes  on,  and  S  nails  in  each 
f^hoe,  at  1  cent  for  the  first  nail,  2  for  the  second,  4  for  the 
third,  and  so  on  ;  v,^hat  was  the  price  of  the  horse  ? 

522.  Sold  30  acres  of  land;  at  2  hob-nails  for  the  first  acre, 
G  for  tiie  second,  18  for  the  third,  and  so  on  ;  and  sold  the 
nails  for  a  farthing  per  100  ;  how  sraicii  did  I  gain,  if  I  gave 
L50  per  acre  for  tlie  land  ? 

52.^.  Mixed  20  bushels  of  wheat  at  5s.,  S6  of  rye  at  3s., 
und  40  of  barley  at  2s. ;  how  much  i:^  a  busliel  of  the  mix^ 
tiire  worth  ? 


184  (luestions, 

624.  A  merchant  would  mix  wines,  at  17s.,  18s.,  and  22s. 
per  gallon,  so  that  the  mixture  may  be  worth  20s.  a  gallon; 
what  quantity  of  each  must  be  taken  ? 

525.  It  is  required  to  mix  brandy  at  8s.,  wine  at  7s.,  cider 
at  Is.,  and  water  at  0  per  gallon,  so  that  the  mixture  may  be 
worth  5s.  per  gallon  ;  what  quantity  of  each  must  be  taken  ? 

526.  What  number  is  that,  which  being  increased  by  i,  i, 
and  I  of  itself,  the  sum  will  be  125.^ 

527.  What  number  is  that,  which  beingmultiplied  by  7, 
and  the  product  divided  by  6,  the  quotient  will  be  14  ? 

528.  One  being  asked  his  age,  said,  if  |  of  the  years  I  have 
lived  be  multiplied  by  7,  and  |  of  them  be  added  to  the  pro- 
duct, the  sum  will  be  292  ;  how  old  was  he  r 

529.  A  son  aking  his  father  how  old  he  was,  received  this 
answer  :  Your  age  is  now  one  fourth  of  mine  ;  but  5  years 
ago,  it  was  only  one  fifth  of  mine.     What  was  the  son's  age? 

530.  How  many  days  can  9  persons  be  placed  in  a  diffe- 
rent position  at  dinner  ? 

531.  How  many  different  ways  can  the  6  vowels  be  put 
together  ? 

532.  How  many  combinations  can  be  made  of  5  letters 
out  of  20  ? 

533.  Tell  the  interest  of  ^365-123,  for  7'S  years,  at  5  per 
cent. 

534.  Of  ^325-5,  for  S'5  years,  at  6  per  cent. 

535.  Tell  the  amount  of  jL35-7,  in  6*2^  years,  at  7'5  per 
cent. 

536.  Of  ^123-65,  in  10  years,  at  6}  percent. 

537.  W^hat  principal  will  amount  to  £565,  in  4  years,  at 
5  per  cent  ? 

538.  What  is  the  piesent  worth  of  ^^390,  payable  in  5 
years,  at  6  per  cent  discount  ? 

539.  At  what  rate  per  cent  will  ^300  amount  to  L428*25, 
in  9'5  years  ? 

540.  In  what  time  will  jL4C0  amount  to  i491,  at  6*5  per 
cent? 

541.  In  what  time  will  ^525  amount  to  X603-75,  at  5  per 
cent  ? 

542.  What  is  the  amount  of  L2S0,  lor  3  years,  at  6  per 
cent,  compound  interest  ? 

543.  W  hat  is  the  compound  interest  of  ^350,  for  5  years, 
at  4  per  cent  ? 

544.  What  principal,  at  5  per  cent,  compoand  interest, 
f©r  4  jears,  will  amount  to  D972'4©5  ? 


^nest. 


185 


545.  WKat  principal,  at  4  per  cent,  compound  interest 
for  5  years,  will  amount  to  L6691-5909632  '? 

546.  In  what  time  will  d^  17500  amount  toL22334-927343r5 
at  5  per  cent,  compound  interest? 

547.  At  what  rate  per  cent  will  ^225   amount  to  X263 
•218176,  in  4  years? 

548.  At  what  rate  per  cent  will  L1234  amount  to  L1469 
•713744,  in  3  years? 

549.  What  is  the  amount  of  an  annuity  of  D600,  for  5 
vears,  allowing  simple  interest  at  b  per  cent  ? 

550.  What  is  the  amount  of  an  annuity  of  D 5 00,  for  4^ 
years,  allowing  simple  interest  at  7  per  cent  ? 

551.  Find  the  present  worth  of  an  annuity  of  D200,  to 
continue  4  years,  at  5  per  cent,  simple  interest. 

552.  Find  the  amount  of  an  annuity  of  D 100,  for  4  years, 
allowing  compound  interest  at  6  per  cent.       < 

553.  What  is  the  amount  of  an  annuity  of  L200,  for  3 
years,  allowing  compound  interest  at  4  per  cent? 

554.  What  is  the  present  worth  of  an  annuity  of  Z.  100,  to 
continue  4  years,  discount  at  4  per  cent,  compound  interest  ? 

555.  Find  the  present  worth  of  an  annuity  of  D300,  to 
continue  3  years,  discount  at  6  per  cent,   compound  interest^ 

556.  Find  the  present  worth  of  a  perpetuity  of  DlOO,  dis- 
counting at  4  per  cent,  compound  interest. 

557.  What  must  i  give  for  an  annuity  of  L50,  to  continue 
forever,  discounting  at  6  per  cent,  compound  interest  ? 

558.  What  perpetuity  can  be  purchased  for  D 1 200,  al- 
lov/ing  discount  at  5  per  cent,  compound  interest? 

559.  What  must  I  give  for  a  perpetuity  of  D80,  to  com- 
mence 4  years  hence,  discounting  at  4  per  cent,  com.  int.  ? 

560.  Which  is  the  most  valuable,  and  how  much  so,  au 
annuity  of  D200  a  year  for  10  years,  or  a  perpetuity  of  D400, 
to  commence  lO  years  hence,  discounting  at  4  percent, 
compound  interest  ? 

561.  Find  the  area  of  a  triangular  field,  one  side  of  which 
measures  186  poles,  and  the  perpendicular  upon  it,  from  the 
opposite  angle,  24  poles. 

562.  Find  the  area  of  a  triangular  field,  the  three  sides  of 
which  measure  126  poles,  100  poles,  and  86  poles. 

563.  Find  the  area  of  a  field  of  six  sides,  which  being  di- 
\Tided  into  4  triangles,  the  bases  and  perpendiculars  measure 
as  follows  : 

triangles,      bases.         perpend,         I      triangles,      bases,         perpend^ 
No.  1        48  poles,    14  poles,  No.  3        47  poles,    18  poles, 

2        35  13  4        80  10 


186  (:luestiGns, 

564.  What  is  the  area  of  a  circle,  whose  dianaeter  is  10 
poles  ? 

665.  What  is  the  circumference  of  a  circle,  whose  diame- 
ter is  25  rods  ? 

566.  Find  the  diameter  of  a  circle,  whose  circumference  is 
£78  rods. 

567.  If  a  piece  of  ground  of  7  sides,  one  of  which  is  78 
rods,  contains  12  acies,  what  will  be  the  content  of  another 
piece  of  the  same  shape,  the  similar  side  ut  which  is  25  rods  ? 

668.  if  the  diameter  of  one  circle  is  20  rods,  what  must  be 
the  diameter  of  another,  to  contain  S  times  as  m.uch  ground  ? 

569.  What  is  the  height  of  a  tree,  when,  it  jou  set  up  a 
perpendicular  pole  'zX)  feet  above  ground,  and  take  such  a 
suUoii  that  }our  eje  is?  iii  a  ran^e  with  the  top  of  the  tree 
and  the  top  ot  tl»e  |julf,  a  our  eye  is  6  feet  from  the  ground, 
and  your  station  10  tVet  hum  the  pole,  and  6'  from  the  tree  ? 

67t.  What  \h  the  breathh  of  a  river,  accoroing  to  Problem 
7s  of  Mensuration,  when  Eb  is  15  rods,  FD  35  rods,  and  DA 
135  rods? 

571.  Find  the  number  of  solid  feet  is  a  stick  of  squared 
timber,  which  measure;-  at  Oie  butt  end,  l6  inches  by  14, 
and  at  the  small  end,  14  by  10,  and  is  45  feet  long? 

.^7'2.  Find  the  superficial  content  of  a  pyramid,  the  base  of 
v/hich  is  1^  feet  scpiare,  and  the  slant  height  25  feet. 

57 S.  Find  the  solid  content  of  a  cone,  the  diameter  of 
whose  base  is  10  teet,  and  the  height  30  feet. 

574.  if  a  cone,  the  diameter  of  whose  base  is  6  feet,  con- 
tains 150  solid  feet,  what  must  be  the  diameter  of  the  base  of 
a  similar  one,  that  shall  contain  300  feet  ? 

575.  What  is  the  solid  content  of  a  stick  of  round  timber, 
■whose  diameter  is  15  inches,  and  the  length  50  teet,  the  dia- 
meter being  the  same  throughout  ? 

576.  What  is  the  superficial  content  of  a  globe,  whose 
diameter  is  25  inches  ? 

577.  In  600/.  Canada,  how  much  New-York  ? 

578.  Reduce  -^}^  to  its  equivalent  nun»ber. 

579.  V^  hat  cost  7 100,  at  ef  d.  each  ? 

5 SO.  Tell  the  least  common  multiple  of  7,  8,  9,  and  10. 

581.  Reduce  |f|  to  its  lowest  terms. 

582.  Reduce  |  of  an  inch  to  the  fraction  of  a  yard. 

583.  Reduce  |  ot  an  English  guine  i  to  sterling. 
554.  In  40/.  sterling,  how  much  Kentucky  ? 
5^§.  T^U  ^le  value  ©ft  ©f  a  £. 


^msttons^  i97 

586.  In  400  f,  sterling,  how  many  livresr 

587.  Multiply  2-714  by  100, 
^88.  Multiply  -3  by  -3. 

589.  Divide  -6  by  6, 

590.  Add  f ,  f,  and  4. 

591.  Tell  the  superficial  content  of  a  globe,  whose  circum- 
ference is  6  feet, 

592.  Tell  the  number  of  solid  inches  in  a  globe,  whose 
diameter  is  3  feet. 

593.  What  is  the  number  of  wine  gallons  which  a  cask 
will  contain,  of  which  the  head  diameter  is  36  inches,  the 
bung  diameter  40  inches,  and  the  length  06  inches  ? 

594.  Find  the  number  of  tons  a  ship  will  carry,  whf  n  the 
length  of  the  keel  is  150  feet,  and  the  breadth  of  the  beam  80 
feet. 

595.  How  many  solid  inches  in  an  irregular  stone,  which 
being  put  into  a  cubical  vessel,  and  the  vessel  filled  with 
water,  and  the  stone  being  taken  out,  the  space  left  empty 
measures  15  inches  long,  l5  wide,  and  4  deep? 

596.  If  a  man  weighing  100  lbs.  rest  on  the  end  of  a  lever, 
12  feet  from  the  prop,  what  weight  wdl  he  balance  at  the 
other  end  of  it,  8  inches  from  the  prop,  no 'allowance  being 
made  for  the  weight  of  the  lever  ? 

597.  If  a  power  weighing  50  lb.  be  applied  to  the  end  of  a 
rope,  round  a  wheel  whose  diameter  is  4  feet,  what  weight 
will  it  balance  suspended  to  a  rope  which  goes  round  an  axle 
6  inches  in  diameter,  no  allowance  being  made  for  friction  ? 

598.  What  is  the  height  of  a  steeple,  from  the  top  of  which 
a  bullet  being  let  fall,  strikes  the  ground  in  4  seconds  ? 

599.  Salisbury  steeple,  in  England,  is  supposed  to  be  400 
feet  high  ;  how  long  w^ould  it  take  a  bullet  to  fall  from  its 
top  to  the  ground  ? 

600.  How  deep  is  a  chasm,  into  which,  if  you  drop  a  stone, 
it  is  8  seconds  before  you  hear  it  strike  the  bottom  ? 

601.  The  population  of  North  America  is  estimated  as 
follows  :  United  States  9640000,  British  possessions  1420000, 
Indians  in  both  510000,  Floridas  25000,  Mexico  75410000, 
Guatimala  1200000,  British  West  Indies  SiOOOO,  Spanish 
do.  900000,  French  150000,  Dutch  80000,  Danish  40000, 
Hayti  600000  ;  how  many  in  all  ? 

602.  Of  South  America  as  follows:  New- Grenada  1 600000, 
Caraccas  900000,  Peru  1500000,  Chili  900900,  Buenos  Ayres 
1100000,  Portuguese  Brazil  2000000,  GuiaEa  50000©,  In- 


188  ^iiessions, 

dians  in  Brazil  2000000,  in  Amazonia  3000000,  in  Fatogo- 
nia,  ^c.  i,iG  000 ;  how  many  in  all  ? 

(303.  Of  Europe  asfoaows:  Great  Britain  and  Ireland 
1 6816^00,  Sweden  i  77000,  Norwuj  i^i^Oou,  Denmark 
164o000,  Russia  3;>829000,  Prussia  9737000,  Holland 
20n2:.00,  Is ethedands  4140000,  France  267750'. 0,  Austria 
£4646j() ',  German  s^tates  150000  0,  Switzerland  1768000, 
Spain  10396000,  Portugal  3559000,  Italy  l6n7Oo0,  Turkej 
988200©,  various  islands  275ooo  ;  how  many  in  all  ? 

604.  Of  Asia  as  follows:  Russia  14  millions,  Turkey  If, 
Persia  ^2,  Arabia  11,  Hindoostan  loo,  Burmah  :2o,  Siam, 
Malacca  and  Lacs  lo,  Japan  3o,  Chinese  empire  !si-o,  Indian 
Archipelago  icio,  Independent  Tartary  2,  Australasia  5,  Po- 
lynesia ?,  Ceylon  and  other  islands  2  ;  how  many  in  all  ? 

605.  The  population  of  Africa  is  estimated  at  98945ooo,  of 
which  the  empire  of  Morocco  is  said  to  contain  '4886ooo, 
Algiers  15ooooo,  Tunis  3  mdlions,  Tripoli  1  million,  Egypt 
3500000,  Nubia  25ooooo,  Abyssinia  2  m  llions,  Madagascar 
4  millions,  British  colonies  ISo-ooo,  British,  Spanish  and  Por- 
tuguese islands  oooooo  ;  and  the  rest  are  savages  ;  how  many 
^re  th    last? 

606.  The  seven  most  populous  cities  of  Asia,  are  Pekin, 
containing  3  millions  ;  Nankin,  2  millions  ;  Canton,  2  mil- 
lions;  Fo-han,  Hang-tchau,  King-te  chmg,  and  Jeddo,  each 
1  million :  how  many  people  in  these  seven  cities? 

t)07.  The  next  ten  cities  in  Asia,  are,  Calcutta,  containing 
650000  inhabitants,  Surat  6ooo©o,  Miaco  5^9726,  Benares 
500000,  Patna  5ooooo,  Susa  5ooooo,  Ispahan  4ooooo,  Madras 
Sooooo,  Erzerum  27oooo,  and  Aleppo  25oooo ;  how  many 
less  in  these  ten,  than  in  the  foregoing  seven  ? 

608.  The  seven  most  populous  cities  of  Europe,  are,  Lon- 
don, containing  1  £00000  inhabitants,  Paris  715595,  Constan- 
tinople 500000,  Naples  412439,  Lisbon  S5oooo,  Moscow 
300000,  and  Fetersburgh  271137 ;  how  many  less  in  these, 
than  in  the  seven  first  cities  of  Asia  ? 

609.  The  next  ten,  are,  Vienna,  containing  252o49,  Am- 
sterdam 217o24,  Dublin  l9oooo,  Berlin  169ooo^  Madrid 
156672,  Palermo  l5oooo,  Barcelona  147ooo,  Ediiiburgh  and 
its  port  138235,  Venice  13724o,  and  Rome  I29ooo;  how 
many  less  in  these  ten,  than  in  the  preceding  seven  ? 

610.  The  seven  most  populous  cities  of  Africa,  are,  Fez, 
containing  38oooo  inhabitants,  Cairo  3ooooo,  Morocco  v7oooo, 
Tanis  150000,  Mequinez  11 0000,  Sennaar  looooo,  and  AI- 


Questions.  1  &9 

pers  80000  ;  how  many  less  in  these,  tlian  in  the  seven  first 
cities  of  Europe  ? 

61 1.  The  seven  most  populous  cities  in  America,  are,  Rio 
Janeiro,  containing  15oooo  inhabitants,  Mexico  ISrooo,  New^ 
York  143706,  Philadelphia  118630,  St.  Salvador  ilOooo, 
Potosi  1 00000,  and  Buenos  Ayres  7oooo  ;  how  many  less  in 
these,  than  in  the  seven  first  cities  of  Africa? 

612.  The  Netherlands  contain  i75oo  square  miles,  and 
41 4o255  inhabitants  ;  how  many  is  that  to  a  square  niile  ? 

613.  The  Italian  possessions  of  Austria  contain  17453 
square  miles,  and  S82ol28  inhabitants;  how  many  is  that  to 
a  square  mile  ? 

6i4.  The  earth  contains  199  millions  of  square  miles,  of 
which  I60  millions  are  sea  and  parts  unknown;  how  many 
square  miles  compose  the  habitable  world  ? 

615.  Of  the  habitable  world,  America  contains  m,  Asia 
^f f ,  Africa  3^^,  Europe  3Y0,  and  New-Holland  the  rest ; 
how  many  square  miles  in  the  last  ? 

616.  According  to  Melish's  map  of  the  United  States,  the 
number  of  square  miles  within  their  limits,  from  the  Atlantic 
to  the  Pacific,  is  2256955.  If  the  population  should  increase 
to  the  amount  mentioned  in  No.  425,  how  many  will  it  be  to 
a  square  mile  ? 

617.  According  to  the  map  of  the  land  of  Canaan,  as  di- 
vided by  Joshua,  it  contained  8362  square  miles ;  from 
which,  deduct  the  territory  of  the  Sidonians,  5o  miSes  long 
and  8  broad,  and  that  of  the  Philistines,  4o  long  &  15  broad  ; 
and  how  many  square  miles  were  left  for  the  Israelites  ? 

618.  The  Israelites,  in  the  time  of  SoloBion,  are  supposed 
to  have  been  seven  millions ;  if  so,  how  many  is  that  to  a 
square  mile  ? 

619.  In  l§ol,  the  population  of  Great  Britain  was  stated 
as  follows :  England  8331434,  Wales  541546,  Scotland 
1599o68,  Army  and  Navy  47o598  ;  how  many  in  all  ? 

620.  In  181  l,as  follows:  England 94994oo,  WaJes  6o738o, 
Scotland  18o4864,Army  &  Navy  64o5oo;  how  many  in  all? 

621.  What  was  the  increase  per  cent  in  those  Jo  ytars? 

622.  Allowinj*:  the  same  increase  per  cent  for  the  next  ten 
years,  what  would  be  the  population  in  1821  ? 

623.  If  England  and  W  ales  had  increased  at  that  rate  per 
cent,  what  would  their  proportion  be  in  182 1  ?  ' 


1 9o  (^iiestions. 

624.  It  is 'estimated  that  there  are,  in  England  and  Wales, 
I0434  Episcopal  clergymen,  and  as  many  dissenters.  If  the 
whole  population  were  equally  divided  among  them  all,  how 
many  souls  would  constitute  the  charge  of  each  minister  ? 

Gis.  The  number  of  Synods,  Presbyteries,  and  Ministers 
in  the  established  church  in  Scotland,  in  18o3,  were  as  fol- 
lows : 

SvnGcls.'                   Presb.  Min. 

Lotiiian  and  Tweeckle,  7  116 

Merse  and  Teviotdaie,  6  66 

Dumfries,  5  54 

Galloway,  3  m 

Glasgow  and  Ayr,  7  130 

Perth  and  SterliDg,  5  SO 

Fife,  4  71 

Angus  and  Mcarns,  G  81 

How  many  Presbyteries,  and  how  many  ministers  in  all  ? 

626.  Suppose  there  were  loo  Burgher  ministers,  So  Anti- 
burghers,  and  £5o  of  other  denominations;  and  add  the  mi- 
nisters of  the  established  church,  and  divide  among  them  the 
whole  population  of  Scotland,  as  it  was  in  I80I,  (See  No. 
619  ;)  and  how  many  souls  will  be  to  each  ? 

62r.  The  number  of  Synods,  Classes,  and  Ministers,  in 
the  established  cliurch  in  Holland,  in  i8o3,  was  as  follows 


S 1/710  ds. 
Aberdeen, 
MorAy, 
Ross, 

Sutherland  &  Caithness, 
Ar^ie, 
Glenelp:, 
Orkney, 


Presb, 

Min. 

8 

101 

■     7 

54. 

3 

23 

;s3,    3 

23 

5 

41 

5 

•  29 

4 

SO 

Synods.  Classesi-        JlLi. 


Synods.  Classes.      JS'lin. 


Gueideriund, 

9 

245 

Friesiand,                         6 

207 

South  Holla.id, 

11 

331 

Overyssel,                        4 

8* 

North  Holland, 

G 

220 

Drente,                             3 

40 

Zealand, 

4 

163 

Groningen,                       7 

161 

Utrecht, 

3 

79 

On  the  i;land  of  Ameland, 

0 

How  many  Classes,  and  how^  many  Mini-lers  in  all? 

628.  There  were  also,  of  oti  er  denominations,  as  follows  : 
Walloon  Calvhiists,  5o  ministers  ;  Kn*.';ii-;h  Presbyterians,  7; 
Scotch  do.  1  ;  P^piscopalians,  2  ;  Catholics,  4oo  {  Lutherans 
7o;  Remonstrants,  ^5  ;  Baptists,  25  i;  Khinsburghers,  So  ; 
Armenians,  I  :  how  many  in  all  ? 

6!i9.  Deduct  50000  Jews  from  the  population  of  Holland, 
as  stated  in  No.  6o3,  and  divide  the  reominder  among  all  the 
ministers  ;  and  how  many  souls,  will  there  be  to  each  ? 

6.  o.  Of  the  population  of  Russia  in  819,  it  was  stated 
that  -.1262000  were  ot  the  established  church  ;  and  the  num- 
ber of  clergymen  in  that  church  in  lHo5,  was  stated  as  fol- 
lows :  Protoires,  Prieijvts,  and  Deacons,  441^7;  Readers  and 
Sacristans,  64'?39.  If  this  population  w^ere  divided  equally 
among  all  the  clergy,  how  many  souls  would  there  be  to  each  ? 


({iiestions.  191 

GS  1.  The  number  of  Methodists  in  18o5,  was  as  follows  : 
in  Great  Britain  lol9l5,  Ireland  23321,  Gibraltar  4o,  West 
Indies  2265o,  America    o2328  ;  how  many  in  all  ?' 

632.  In  1819,  the  number  in  the  United  States  was  as 
follows  :  Ohio  Conference  29  i  34,  Missouri  4764,  Tennessee 
2o6r6,  Mississippi  ^'>71,  S.  Carolina  32646,  Yirginia  22585, 
Baltimore  3^7:6,  Philadelphia  32796,  New-York  22638, 
New-Encrland  15312^  Genesee  23913:  how  many  in  all? 

633.  Tiie  number  of  travelling  preachers  was  812;  how 
many  were  under  tlie  care  of  each,  on  an  average  ? 

634.  The  number  of  local  preachers  was  staged  to  be  more 
than  looo  ;  suppose  them  to  be  1 188  ;  and  how  many  mem- 
bers would  there  be  to  a  preacher  ? 

635.  The  same  Tear,  the  number  under  the  care  of  the 
British  and  Irish  Conferences,  was  242459  ;  how  many  were 
in  all  the  world  ? 

656.  The  following  statement  of  the  number  of  Baptists 
in  the  United  States,  is  compilfed  from  the  returns  made  to 
their  Gen.  Convention  in  1 82 1.  Wliere  there  were  blanks  in 
those  returns,  they  are  filled  by  estimates,  and  marked  with 
an  asterisk.  The  number  in  New-England  is  stated  as  fol- 
lows : 


min. 

C/.A-9. 

inem. 

min. 

chhs. 

mem.. 

Maine, 

129 

171 

9740 

Massachusetts, 

105 

109 

10078 

N.  Hampshire, 

34 

44 

2765 

Rh.  Island, 

41 

54 

r.052 

Vermont, 

97 

121 

9978 

Connecticut, 

59 

60 

6946 

How  many  ministers,  how  many  churches,  and  how  many 
i.x3mbers? 

637.  In  the  Middle  states,  as  follows : 

f  New-Y(irk,  304        426       35200 .Delaware,  7  7  55S 

j   N.Jersey,  22  23        £225  :vL,rylar.d,  19         36  $06 

j   *Pemuylvan!a,       76  81         597r)IColumbla  Dist.  11         16         1511 

].       How  many  ministers,  how  many  churches,  and  how  many 
'members'? 

638.  In  the  Southern  states,  as  follows  : 

»\irginisi,  157         300       21000  Geortria,  101         181        1457^ 

*N.  Carolina,       198        233       11924!"A'abama,  42  61         252? 

*:5.  Carohua,  95         190       14401  ^Mississippi,  36  68        2006 

How  many  ministers,  how  many  churches,  and  how  many 
members  ? 

639    In  the  Western  states,  as  follows  : 


*rennessee. 

123 

170 

9707 

*ln(liana. 

61 

102 

3886 

^Kentucky, 

240 

395 

27299 

tllinois, 

19 

15 

332 

*Ohio, 

87 

150 

5557 

*  Missouri, 

27 

37 

726 

\ 


How  many  ministers,  how  many  churches,  and  how  many 
members  ? 


1§2  Questions, 

64.0.  Take  the  total  of  the  ministers,  churches,  and  mem- 
bers, in  the  preceding  four  que-^tioos,  and  add  the  seventh 
day  Baptists,  esti'-iateil  at  15  in'misters,  2o  churches,  and 
2ooo  members  ;  and  what  is  the  amount  ? 

64}.   tlow  many  more  churches  than  ministers? 

64^^'.  In  l^-it  J  the  General  Assembly  of  the  Presbyterian 
Church  in  the  United'  States,  had  under  iis  care  ^2  i^.v.>by- 
ieries,  of  which  5o  reported  l3oo  co  igregations,  7 ''4  or  iain- 
ed,  and  lo3  licensed  preachers;  if  those  wliich  .iui  not  re- 
port, contained  each  two  thirds  as  many  in  proportion  as 
those  which  did,  how  many  congregations,  and  how  many 
preachers  in  kll  ? 

643.  If  every  preacher  supplied  a  congregation,  how  many 
congregations  v/ere  <lestitute? 

644.  in  65  1  coiigregations,  the  number  in  communion  was 
71364  ;  what  is  the  average  to  each  ? 

645.  If  the  i'est  consained  an  average  of  the  same  number 
each,  what  would  be  the  whole  number  of  communicants  in 
that  connection  ? 

646.  [n  l8>2o,  the  several  Congregational  associations  of 
Kew- Hampshire,  reported  as  follows  : 


min. 

c'^hs. 

mm. 

chh^. 

Deeifild, 

8 

9 

Orange, 

8 

12 

Haveihill, 

6 

8 

Pisc^Uqua, 

IG 

15 

Holies, 

4 

5 

Plvtnouth, 

3 

6 

Hopkiuton, 

11 

12 

Union, 

7 

0 

Mofiadiock, 

14 

17 

Coos. 

3 

10 

If  there  are  lo  ministers  and  lo  churches  not  connected 
v/ith  these  associations,  and  8  licensed  preachers  ;  what  is 
the  whole  number  of  preachers  and  churches  of  this  order  ia 
New  Hampshire  / 

647.  In  81  of  these  churches,  there  were  8843  members; 
if  the  rest. averaged  the  same  number  each,  what  was  the 
whole  number  ot  members  ? 

648.  In  i8lo,  the  Congregational  ministers  and  licentiates 
in  the  several  associations  in  Vermont,  were  as  follows  : 

Windham,  13  0  |         Royaltoa,  11  0 

Rutland,  19  7  Orange,  13  5 

Adlison,  13  2         j         N.  Wester*,  8  0 

How  many  in  all  ? 

6;9.  In  18-4,  the  number  was  79  settled  ministers,  1© 
unsettled,  &  8  licentiates;  what  was  the  increase  in  4years 

65o.  At  the  same  rate  of  increase,  what  would  be  the 
BLUmber  ia  l82o  ? 


((uestwns.  19S 

651.  If  there  were  140  churches  of  109  members  in  each;, 
how  many  members  were  there  in  all  ? 

^^5^.  la  18:0,  the  ministers  and  churches  connected  with 
the  Congregational  Associations  of  Connecticut,  were  as 
follows : 


min. 

cJihs. 

min. 

chhs.. 

Ilartfoid  North, 

21 

20 

Fairfield  East, 

8 

VZ 

Haitford  South, 

16 

16 

V^indham, 

20 

2$ 

New-Haven  West, 

16 

19 

Litchfitld  North, 

18 

18 

New   Haven  East, 

11 

1.'3 

Litchfield  South, 

15 

18 

New-London, 

15 

21 

Middlesex, 

15 

15 

Fairfield  Wt^st, 

10 

i; 

lolland. 

15 

15 

Besides  which,  there  were  2  1  unsettled  ministers,  and  9 
licensed  preachers  ;  how  many  ministers,  how  manj  church- 
es, and  how  many  members,  if  the  churches  average  lOy  each? 

653.  The  number  of  settled  Congregational  ministers  in 
Massachusetts,  was  stated,  in  816,  to  be  3'23.  Suppose 
thefn,  in  :820,  to  be  330,  and  the  number  of  vacant  churches 
to  be  5o  ;  suppose  the  unsettled  ministers  and  licentiates  to 
be  ^0,  and  the  churches  to  average  1*  9  members  ;  and  what 
is  the  number  of  Congregational  ministers,  of  churches,  and 
of  members,  in  Massachusetts  ? 

654.  The  number  of  settled  Congregational  ministers  in 
lihode -Island,  was  stated,  in  1816,  to  be  b.  Suppose  them, 
in  1820,  to  be  9,  and  the  number  of  vacant  churches  to  be  4  ; 
supj>ose  the  unsettled  ministers  and  licentiates  to  be  3,  and 
the  churches  to  average  I'^t'  members  ;  and  what  is  the  num« 
ber  of  ministers,  of  churches,  and  of  members,  in  Rhode- 
Island  ? 

65  -.  Suppose  t^  ere  are,  in  the  states  out  of  New-England, 
110  Congregational  ministers,  and  150  churches  of  i09  mem- 
bers each,  and  in  Maine  the  same  as  in  Vermont ;  and  what 
is  the  whole  number  of  ministers^  churches,  and  members,  m 
the  United  States  r 

656.  The  number  of  ministers  and  churches  of  the  Asso- 
ciate Reformed  Presbyterians,  in  18  16,  was  as  follows  : 
Synod  of  New-York,  2  Min.  32  Chhs.;  Pennsylvania,  9 
IMin.  23  Chl»s. ;  Sciota,  23  Min.  53  Chhs. :  how  many  minis- 
ters, how  many  churches,  and  how  many  members,  if  thej 
average  at  109  members  each  ? 

6:*.  Suppose  the  Associate  Presbyterians  to  be  20  minis- 
ters  and  3o  churches,  and  the  Cumberland  Presbyterians  to 
be  25  ministers  and  40  churches,  and  that  these  churches 
av^erage  109  members  each  ;  and  how  many  ministers., 
churches,  and  members,  in  these  two  bodies  ? 

H 


194  %iestions, 

658.  The  Dutch  Tleformed  Church,  in  1 820, contained  tht 
following  classes,  ministers,  and  churches  : 


Classes  of 

min. 

chhs. 

Classes  of 

min. 

chhs. 

Nev.-Yoi'k, 

16 

15 

Albtniy, 

10 

16 

New-Brunswiek, 

9 

10 

Washington, 

4 

9 

Bergtn, 

9 

13 

Poughkeepsie, 

8 

11 

*Paramus, 

8 

11 

Ulster, 

8 

20 

*Long-lsland, 

7 

9 

*  Montgomery, 

11 

14 

Pliiladelphia, 

7 

6 

*Rensselaer, 

5 

6 

«^^7^.  emig. 

In  Pennsylvania,             50  246 

Ohio,                                5  52 

>taiylancl,                      7  34 

Virginia,                          4  33 


Besides  which,  there  were  4  licentiates  reported,  and  pro- 
bably as  many  more  not  reported ;  if  so,  how  many  ministers, 
and  churches,  in  that  body  ? 

659.  Of  these  churches,  67  reported  902S  communicants  ; 
if  the  rest  averaged  1(9  each,  what  is  the  whole  number  ? 

66i;.  The  German  Reformed  Church,  in  1821,  reported  as 
follows  : 

min.     cong. 

N.  Carolina,      .  1  28 

S.  Carolina,  0  8 

Tennessee,  Kentncky,  &  elsewhere, 

(estimated,)  '     ^13  99 

Besides  which,  there  were  10  licentiates;  how  many  mi- 
nisters, how  many  congregations,  and  how  many  communi- 
cants, if  the  congregations  average  30  each  ? 

66 1.  Add  the  ministers,  churches,  and  members,  as  above 
stated,  of  the  General  Assembly,  the  Congregationalists,  the 
Associate  Reformed,  the  Associate  and  Cumberland  Presby- 
terians, the  Dutch  anu  German  Reformed  Churches;  and  find 
how  many  ministers,  how  many  churches,  and  how  many 
members  in  communion. 

662,  The  number  of  Episcopal  clergymen  in  the  United 
States  in  'v8l7,  was  stated  as  Ibllows  :  New-'flampshirp,  4  y 
Massachusetts,  13;  Vermont,  4;  Rhode-Island,  4;  Con- 
necticut, 35;  NeW'York,  67;  New-Jersey,  11;  Pennsyl- 
vania, 25  ;  Delaware,  3;  Maryland,  36;  Virginia,  33  ;  N. 
Carolina,  3  ;  S.  Carolina,  17 ;  how  many  in  all  r 

(563.  In  1821,  there  were  9  bishops,  200  presbyters,  and 
124  deacons;  what  was  the  increase  in  4  years  ? 

664.  Of  these,  there  were,  in  the  diocese  of  New-York, 
77  ;  Maryland,  5  '■  ;  Virginia,  33  ;  Ohio,  5  :  how  many  in  the 
ether  live  dioceses  ? 

665.  If  there  are  500  churches,  with  80  communicants  in 
each,  what  is  the  whole  number  ? 

666.  If  the  number  of  Lutheraii  ministers  is  100,  and  their 
churches  150,  with  80  members  in  each;  what  is  the  number 
of  their  communicants  ? 

*  EstbiiRtes, 


Questions.  195 

667.  What  is  ihc  whole  number  of  preachers  of  all  the 
jove  denonnnations,  according  to  Nos.  633,  634,  640,  661, 

663,  and  666  ? 

668.  If  500  of  these  ministers  are  employed  as  officers  of 
colleges,  instructors  of  academies,  or  are,  from  various  causes, 
not  employed  as  ministers  ;  and  the  rest,  except  the  Metho- 
dists, are  all  engaged  in  supplying  each  one  church  ;  how 
many  churches  are  destitute? 

669.  How  many  more  preachers  are  wanted,  that  every 
800  souls  in  the  United  States  may  have  one  to  supply  them? 

670.  What  is  the  whole  number  of  communicants  of  all 
ihii  above  denominations  ? 

671.  How  many  are  not  communicants  of  any  of  these 
denominations,  for  one  that  is  ? 

67-2.  The  expenditures  of  the  Worcester  county  charitable 
society  in  1817,  were,  for  education,  S703*48  ;  foreign  mis- 
sions, $2i)4-9^  ;  feeble  churches,  $280  ;  bibles,  &c.  $57*45  ; 
ontingent  expenses,  §5*05  :  how  many  guilders  in  the  whole? 

673.  The  expenses  of  the  Deaf  and  Dumb  Asylum  at 
Hartford,  in  the  year  )8l8,  were,  for  buildings  and  lands, 
.^'S860-85  ;  for  tuition,  S3283-87  ;  boai\l  of  pupils,  $8398-80: 
how  many  rubles  in  the  whole  ? 

674.  The  receipts  were,  donations,  g7528'48  ;  paid  by 
pupils, . $5843*20  ;  contributioQS  from  churches  in  Connecti- 
cut, |2546-J  2  ;  interest  of  fund,  $1018-42  :  how  many  mill- 

eas  in  the  wliole  ? 

675.  The  payments  of  the  Female  Society  in  Boston  foi 
he  conversion  of  the  Jews,  in  1818,  were,  for  board,  &c.  of 

N.  Myers,  D40;  sent  the  Society  at  London,  1)444*44;  for 
the  education  of  Jewish  children  at  Bombay,  1)100  ;  printing, 
D23-50;  freight,  D2-35 ;  exchange,  D6.'91  ;  deposited  in 
bank,  D129  :  how  many  marcs  banco  in  the  whole  ? 

676.  In  Liberty  county,  Georgia,  there  was  said  to  have 
been  contributed,  in  i8l8,  for  charitable  and  religious  pur- 
poses, by  75  persons,  asioliows:  For  free  schools,  D 1600  ; 
bible  society,  D20v}0 ;  clergymen,  D3000  ;  female  asylum, 
D650 ;  missionary,  tract,  and  education  societies,  DlOOO. 
if  37  persons  paid  Do  each,  what  w^as  the  average  to  the  rest?. 

677.  A  laboring  man  in  V(r.aont,  saved  the  following 
amount  in  one  year,  for  charitable  purposes :  By  working  on 
the  4th  of  July,  7  5  cents  ;  by  not  wearing  a  cravat,  one  dol- 
lar ;  by  doing  without  ardent  spirits,  one  dollar  ;  by  having 
his  cloth  only  colored,  but  not  dressed,  l)i**;5;  by  wearing, 


196  Questions. 

himself  and  family,  thick   shoes,  4  dollars :  what  was  the 
amount  ? 

678.  If  every  tenth  person  in  the  United  States  would 
"  go  and  do  likewise,"  how  great  a  fund  would  it  annually 
raise  ? 

679.  In  1831,  Rev.  Mr.  Ward  collected  for  the  Baptist 
Mission  College  at  Serampore,  in  New-York,  D2467*19  ; 
Boston,  D]8b0-62;  Philadelphia,  D]202-6!2;  Baltimore, 
D420  ;  Washington,  D2r  1  ;  Princeton,  DM2  ;  New- Haven, 
D406-50;  Hartford,  D28( -06;  Providence,  D3 12-68  ;  Alex- 
andria, D40  ;  Newark,  D9S-1 9;  Pawtucket,  D59  ;  Middle- 
town,  D103;  Schenectady,  Dl90;  Worcester,  Dlb:0-37  ; 
Cambridge,  Dl81  ;  Salem,  0200-72;  Portland,  D24I-06; 
North- Yarmouth,  D85-73  ;  Portsmouth, D84-42  ;  eleven  other 
towns  in  Massachusetts,  D701'04  :  how  much  in  all  ? 

680.  Kean,  the  playactor,  visited  several  of  our  principal 
cities  near  the  same  time,  and  received  for  himself  about 
D50000,  for  his  winter's  wages;  how  many  missionaries 
would  that  support  for  six  months,  at  DsOO  a  year  each  ? 

681.  The  expenses  of  the  Richmond  theatre  for  the  two 
last  seasons,  were  stated,  in  1821',  to  have  exceeded  the  re- 
ceipts by  D 19884  ;  how  many  bibles  would  that  furnish  for 
distribution,  at  60  cents  each  ? 

682.  The  receipts  of  the  Domestic  Missionary  Society  of 
Massachusetts,  for  1820,  were  D6 19-63;  the  balance  on 
hand  from  the  former  year,  was  D380-15  ;  and  the  expendi- 
tures of  that  year  were  D644-48  :  what  was  left  ? 

683.  The  receipts  of  the  Connecticut  Missionary  Society, 
for  1818,  were,  donations  and  interest,  D5052-2I§  ;  contri- 
butions, D32l3'24J  ;  and  the  expenses  wxre  D7244-57 : 
what  was  the  excess  of  expenditure  ? 

684.  The  productive  property  of  the  Massachusetts  Mis- 
sionary Society,  in  1821,  was  D5 327-28  ;  what  is  the  annual 
interest,  at  6  per  cent  .^ 

685..  If  the  expense  of  the  American  Board  is  D57144, 
and  should  be  equally  divided  among  the  inhabitants  of 
viassachusetts,  how  much  would  it  be  for  each  ?  (See  No. 
104.) 

686.  In  1803,  the  Society  in  Scotland  for  propagating  the 
gospel  at  home,  had  sent  out  in  all,  100  missionaries,  and 
had  received  in  all,  X2683 ..  5  ..  1 1,  and  paid  for  tracts, 
1/356  ..0  ..  9  ;  what  was  there  left  for  each  missionary  ? 

687.  In  1793,  there  were  4  missionaries  in  India;  in  1809, 


Questions,  197 

there  had  been  added  5  Episcopalians,  14  Baptists  from  Eu- 
rope, 3  Hindoo  Baptists,  1  Presbyterian,  6  Independents,  2 
Lutherans  ;  besides  which,  there  were  3  missionaries  in 
Ceylon,  and  1  in  China ;  how  many  in  all  ? 

688.  In  a  certain  town  in  the  interior  of  New-England, 
containing  3000  inhabitants,  there  was  raised,  m  1803,  for 
schools,  D800;  the  poor,  DlOOO;  taxes,  D900 ;  support  of 
ministers,  D670 ;  highw^ays,  D3000 ;  incidental  expenses, 
D 1 000  :  how  much  is  that  for  each  inliabitant  ? 

689.  In  the  same  year,  there  were  retailed  in  the  town, 
10230  gals,  of  N.  E.  rum,  at  6i  cents  a  gaL  ;  5900  gals.  W. 
I.  rum,  at  Dl  a  gal. ;  1500  gals,  brandy,  at  Dl'50  a  gal. ;  and 
rso  gals,  gin,  at  1)  1*50  a  gal. :  what  is  the  expense  for  each 

iihabitanf  ? 

690.  What  quantity  of  ardent  spirits  for  each  ? 

691.  In  a  certain  district  of  Bengal,  in  two  months  of 
1812,  70  widows  were  burnt  alive  on  the  funeral  piles  of 
their  deceased  husbands ;  how  many  would  that  be  in  a  year? 

692.  These  70  left  1 84  orphan  children  ;  how  many  were 
left  by  the  whole,  at  the  same  rate  ? 

693.  Within  30  miles  of  Calcutta,  there  were  275  widows 
burnt  alive  in  1803;  if  that  extent  contains  785000  inha- 
jitants,  how  many  would  be  burnt  in  the  whole  of  Hindoos 

ill,  at  the  same  rate  ?     (See  No.  604.) 

6y4.  If  there  are  35000  widows  annually  burnt  alive  in 
Ji'^  whole  of  Hindoostan,  as  is  supposed  to  be  the  fact,  how 
many  orphans  are  thus  annually  made,  at  the  above  rate? 

695.  There  are  12  pilgrimages  annually  made  to  the  single 
ernple  of  Juggernaut  in  Orissa,  at  each  of  which  from  100000 
cO  600000  persons  attend,  of  which  a  vast  proportion  (some 
think  a  large  majority)  never  return  home,  but  die,  from 
want,  fatigue,  fevers,  &,c.  Suppose  the  average  attendance 
to  be  300000,  and  that  of  these,  only  one  in  five  die  ;  what  is 
ihe  number  of  lives  annually -sacrificed  to  this  one  Idol  ? 

69^>.  In  the  wars  kindled  by  the  ambition  of  Bonaparte, 
from  1800  to  1815,  it  is  estimated  that  the  following  lives 
were  lost:  In  Hayti,  160000;  in  the  war  with  England,  du- 
ring 12  years,  200000;  in  the  invasion  of  Egypt,  60000;  in 
the  winter  campaign  of  1805  and  160<S,  one  hundred  and 
fifty  thousand  ;  in  Calabria,  ^  years,  500000 ;  in  the  North, 
in  1806 and  1807,  three  hundred  thousand;* in  Spain,  seven 
years,  2100000;  in  Germany  and  Poland,  in  1809,  three 
hundred  thousand  ;  in  the  invasion  of  Russia,  one  nuHion  ; 

R2 


198  Questions, 

in  the  subsequent  year,  450000  ;  from  his  return  from  Elba, 
to  his  last  dethronement,  ^./0(,'000  :  how  many  lives  were  sa- 
crificecf  to  the  ambitiim  of  that  one  man  ? 

697.  The  Delude  took  place  2348  years  before  the  Chris- 
tian era  :  from  that  to  the  building  of  Babel,  was  101  years  ; 
from  that  to  the  begnning  of  the  kingdom  of  l^gypt  by  Miz- 
raim,  59  years  ;  from  that  to  the  beginning  of  the  kmgdom 
oi  Sieyon,  9:^  yeiirs  ;  from  that  to  the  beginning  of  the  kmg- 
dom  of  Assyria,  30  years  ;  from  that  to  (he  birth  of  Abraham, 
63  years  ;  from  that  to  the  founding  of  Argos  by  Inachus, 
140  years  :  and  from  tliat  to  the  selling  of  Joseph  into  E^ypt, 
128  years ;  in  what  year  did  each  of  these  take  place  ? 

698.  Tlie  kingdom  of  Athenb  was  begun  byCecrops,  1556 
years  before  the  Christian  era;  fr^m  that  to  the  budding  of 
Troy  by  Scamander,  w^as  10  years  ;  fnnn  that  to  the  building 
of  Thebes  by  Cadmus,  53  years  ;  from  tliat  to  the  departure 
of  the  Israelites  from  Egypt,  2  years  ;  from  that  to  the  Ar- 
gonautic  expedition,  228  years  ;  from  tliat  to  the  destruction 
ofTro}^,  79  years  ;  from  that  to  the  building  of  Alba  Lon^a^ 
32  years  ;  from  that  to  Saul's  being  made  king  of  Israel,  37 
years  ;  and  from  that  to  the  death  of  Codrus,  the  last  kin^- 
of  Athens,  25   years  :  in  what  year  diil  each  of  these  take 


ce 


? 


pla 

699.  Solomon's  tem.ple  was  dedicated  1004  years  before 
the  Christian  era,  and  the  kingdom  of  Israel  and  Judah  was 
divided  29  years  after  that,  and  Homer  flourished  68  years 
after  that,  and  Lycurgus  flourished  23  years  after  that,  and 
Carthage  was  built  15  years  after  that,  and  the  first  Assyrian 
empire  was  ended  49  years  after  tlia^,  and  Rome  was  built 
67  years  after  that,  and  the  kingdom  of  Israel  was  ended  52 
years  after  that,  and  Draco  liourislied  98  years  after  that;  lit 
what  year  did  each  of  these  take  place  ? 

700.  Nineveh  was  destroyed  C12  years  before  the  Chris- 
tian era,  and  the  Babylonish  captivity  began  6  years  after 
that,  and  Solon  flourished  15  years  after  that,  and  the  king- 
dom of  Judah  was  ended  4  years  after  that,  and  Babylon  was 
taken  by  Cyrus  49  years  after  that,  and  the  Jews  returned 
from  captivity  2  years  after  that,  and  Confucius  flourished  15 
years  after  that,  and  Rome  became  a  republic  12  years  after 
that,  and  Nehemiah  was  governor  of  Judea  54  years  after 
iliat ;  in  what  year  did  each  of  these  take  place  ? 

7011  Socrates  died  400  years  before  the  Christian  era,  and 
Plato  flourished  12  years  after  that,  and  Aristotle  and  Dc- 


Questions*  199 

mosthenes  flourished  48  years  after  that,  and  Alexander  the 
/Great  began  to  reign  4  years  after  that,  and  died  13  years 
/  after  that,  and  Euclid  flourished  32  years  after  that,  and  the 
•    first  Punic  war  began  27  years  after  that,  and  Archimedes 
flourished  40  years  after  that,  and  the  second  Funic  war  be- 
gan 6  years  after  that,  and  the  battle  of  Cannse  was  2  years 
after  that,  and  Judas  Maccabeus  flourished  50  years  after  that; 
in  what  year  did  each  of  these  take  place  ? 

7{)2.  The  third  Funic  war  began  149  years  before  the 
Christian  era,  and  Carthage  was  destroyed  2  years  after  that, 
the  Jugurthine  war  began  36  years  after  that,  and  Sylla  be- 
came dictator  29  years  after  that,  and  Cicero  flourished  22 
years  after  that,  and  the  battle  of  Pharsalia  was  12  years 
after  that,  and  the  death  of  Csesar  was  4  years  after  that,  and 
Rome-became  an  empire  under  Augustus  13  years  after  that, 
and  our  Saviour  was  born  27  years  after  that ;  in  what  year 
did  each  of  these  take  place  ? 

703.  In  the  year  1813,  the  war  expense  of  Great  Britain 
was  estimated  at  540  millions  of  dollars  ;  France,  and  her 
tributaries,  620  millions  ;  Sweden,  Denmark,  Russia,  Prus- 
slvi,  Austria,  and  their  allies,  800  millions  ;  Spain  and  Por- 
Vj^iil,    150  millions;  United  States,  50  millions:    Spanish 

lonies,  100  millions  :  what  is  the  whole  amount  of  the  ex- 
nse  of  war,  to  these  professedly  Christian  nations,  in  that 
igle  year  ? 

704.  How  many  ministers  of  the  gospel  of  peace,  would 
at  sum  support,  at  $700  each  ? 

705.  If  the  whole  population  of  the  world  should  be  divi- 
•d  equally  among  them,  how  many  souls  would  it  be  to  each? 

:ee  No.  \3.) 

706.  The  report  of  the  Secretary  of  the  Treasury  of  the 
nited  States,  at  the  close  of  the  year  i  8 1 5,  stated  the  pub- 
•  debt  contracted  during   the  last  war  with  Great  Britain, 

to  be  hO  and  a  half  millions  of  dollars;  how  muth  is  that  to 
-  each  individual  of  the  United  States:,  according  to  the  census 
of  1810?     (See  No.  4-08.) 

707.  How  many  bibles  would  it  have  furnished  for  the 
destitute,  at  60  cents  each  ? 

708.  If  you  use  one  tea-spoonful  of  sugar  in  a  cup  of  tea, 
and  drink  6  cups  in  a  day,  and  4  tea-spoonfuls  of  sugar  weigh 
an  ounce,  and  the  sugar  costs  15  cents  a  lb. ;  how  much  can 
you  save  in  a  year,  for  doing  good,  by  drinking  your  tea 
'.vithout  sugar? 

709.  How  many  persons  must  do  without  sugar  m  tkeir 


\ 


200  Qiiestions. 

tea,  that  the  saving  may  maintain  a  Christian  free  school  is 
Ceylon  for  50  heathen  children,  when  such  a  school  costs  2 
dollars  a  month  ? 

7i0.  If  you  smoke  3  cigars  a  day,  and  they  cost  6  cents  a 
dozen  ;  how  many  bibles  would  the  amount  send  to  the  de- 
stitute in  a  year,  at  60  cents  each  ? 

711.  If  a  lady  expends  6  cents  a  week  for  snuff,  how  many 
pages  of  tracts  would  the  amount  pay  for  in  a  year,  at  1  mill 
a  page  ? 

712.  If  a  gentleman  smokes  6  Spanish  cigars  in  a  day,  at  20 
cents  a  dozen,  how  much  might  he  save  in  a  year  for  doing 
good,  by  denying  himself  that  indulgence  ? 

713.  How  many  heathen  children  might  be  educated  by 
that  saving,  in  20  years,  at  the  rate  mentioned  in  Nos.  434 
and  435  ? 

714.  If  a  gentleman  wears  out  3  shirts  in  a  year,  and  the 
additional  expense  of  ruffles  is  75  cents  each  ;  and  if  he  wears 
2  a  week,  and  the  additional  expense  of  washing  and  ironing 
is  2  cents  a  piece  each  time  ;  how  much  can  he  save  in  a  year 
for  doing  good,  by  wearing  plain  shirts  ? 

715.  How  many  heathen  children  would  it  keep  at  school, 
at  48  cents  each  ? 

7i  6.  If  a  child  is  allowed  to  spend  1  cent  a  day,  for  sugar 
plums  and  the  like,  how  many  bibles  would  that  send  to  the 
destitute  annually  ? 

717.  How  many  children  must  make  that  saving,  to  sup- 
port one  orphan  in  the  missionary  family  at  Ceylon,  at  12 
dollars  a  year  ? 

718-  If  a  boy  eats  one  pint  of  nuts  a  week,  at  6  cents  a 
quart,  how  many  boys  must  deny  themselves  that  indulgence, 
that  the  saving  may  support  one  erphan  at  Ceylon  ? 

719.  If  a  chihl  eats  3  apples  a  day,  at  6  cents  a  dozen, 
what  is  the  amount  in  a  year  ? 

720.  How  many  children  must  deny  themselves  this  in- 
dulgence, to  support  a  school  for  50  heathen  children  ? 

721.  If  a  family  make  use  of  2  lb.  of  sweetmeats  a  week, 
worth  20  cents  a  lb.,  how  much  would  they  save  in  a  year  for 
doing  good,  by  denying  themselves  this  luxury  ? 

722.  How  many  tracts,  of  12  pages  each,  would  it  pay  for? 

723.  If  3  tea-spoonfuls  of  sugar  a  day  be  allowed  for  each 
person  in  the  United  States,  according  to  the  census  of  1 820, 
and  the  weight  and  cost  is  as  stated  in  No-  708,  how  much 
might  be  saved  every  year  for  doing  good,  if  all  would  deny 
themselves  this  indulgence  ? 


Questions.  201 

/724.  How  many  missionaries  would  it  support,  at  500 
dollars  each  ? 

725.  How  many  bibles  would  it  pay  for,  at  60  cents  ? 

726.  How  many  young  men  would  it  assist  in  obtaining  an 
education,  at  i25  dollars  each  ? 

727.  How  many  orphans  would  it  support  at  Ceylon  ? 
7£8.  How  many  months  of  the  year  must  the  people  of  the 

United  States  drink  their  tea  without  sugar,  that  the  saving 
may  support  one  minister  of  the  gospel  for  every  800  souls, 
at  600  dollars  each  per  annum  ? 

729.  If  a  man  drinks  half  a  gill  of  ardent  spirits  a  day,  at 
1  dollar  a  gallon,  how  many  tracts,  of  12  pages  each,  would 
the  amount  annually  pay  for? 

7S0.  If  a  man  makes  use  of  half  a  pint  a  day  for  himself 
and  friends,  how  many  bibles  would  that  amount  annually 
pay  for  ? 

73 1^  How  long  must  such  a  man  abstain  from  that  poison, 
that  the  saving  may  furnish  him  with  a  library  of  100  volumes, 
at  1  dollar  50  cents  a  volume  ? 

732.  How  long  to  pay  for  100  acres  of  land,  in  the  new 
settlements,  at  2  dollars  an  acre  ? 

733.  How  many  such  men,  by  abstaining  from  ardent  spi*- 
fits,  could  support  a  minister  of  the  gospel,  at'!iS6<  0  a  year  ? 

734.  In  the  year  1810,  the  quantity  of  ardent  spirits  made 
and  imported  into  the  United  States,  over  and  above  what 
Was  exported,  was  stated  to  be  33365529  gallons  ;  hov/  much 
is  that  for  each  person,  according  to  the  census  of  the  same 
year?     (See  No.  408.) 

735.  What  is  the  expense  to  each  person,  at  1  doll,  a  gal.? 

736.  Supposing  this  quantity  annually  consumed,  how 
long  must  the  people  of  the  United  States  do  without  ardent 
spirits,  that  the  saving  may  supply  the  whole  world  with 
bibles,  allowing  1  bible  to  every  5  persons  ?  (See  No.  13.) 

737.  If  one  minister  of  the  gospel  should  be  allotted  to 
every  SOo  souls,  how  much  of  the  year  must  the  people  of  the 
United  States  do  without  ardent  spirits,  that  the  savirjg  may 
support  them  all,  at  S600  each  per  annum  ? 

738.  How  many  charity  scholars  would  the  expense  of  ar- 
dent spirits  annually  support,  at  ^125  each  per  annum  ? 

739.  How  many  of  our  enterprising  young  men  would  that 
expense  aanually  furnish  with  farms  of  100  acres  each,  at  2 
4o]lars  an  acre  ? 

740.  How  many  persons  would  that  expense  annually  sup- 


502  (luestions- 

ply  with  bread,  allowing  5  bbls.  of  Hour  to  every  6  persons^ 
at  5  dolls,  a  bbl.  ? 

741.  How  many  more  persons  is  that,  than  the  whole  po- 
pulation of  the  United  States  in  I8i0,  by  whom  the  ardent 
spirits  were  consumed  ? 

742.  If  turnpike  road  can  be  made  for  300  dollars  a  mile, 
how  long  must  the  people  of  the  United  States  do  without 
ardent  spirits,  that  the  saving  may  make  a  turnpike  road  that 
would  reach  round  the  globe  ? 

743.  How  long,  to  make  such  a  road  from  Boston  to  the 
mouth  of  the  Columbia  river,  which  is  estimated  to  be  2800 
miles? 

744.  Hov^^  deep  would  the  above  quantity  of  ardent  spirits 
fill  a  pond  of  ten  acres  ? 

745.  How  many  acres  would  it  cover  a  foot  deep  ? 

746.  How  many,  an  inch  deep  ? 

747-  If  it  was  put  up  in  hhds.  of  63  gals,  each,  hov/  man\ 
would  it  fill? 

748.  How  many  waggons  w^ould  it  load,  at  2  hhds.  each  ? 

749.  How  many  miles  would  they  reach,  allowing  3  rods 
to  each  waggon  and  horses  ? 

750.  If  the  annual  expense  of  ardent  spirits  to  the  people 
of  the  United  States,  is  33365529  dollars,  how  much  is  that 
per  minute  all  the  time,  reckoning  365 1  days  to  a  year  ? 

751.  What  is  the  share  of  the  state  of  New- York  in  that 
expense  ?     (See  No.  405.) 

752.  If  the  Great  Erie  Canal  should  bo  360  miles  long,and 
cost  12600  dollars  a  mile,  how  long  must  the  people  of  the 
state  of  New-York  do  without  ardent  spirits,  to  defray  the 
expense  ? 

753.  How  long,  to  endow  an  academy  in  each  of  the  coun- 
ties in  the  state,  (being  then  45,)  with  a  fund,  that,  at  7  per 
cent,  shall  yield  600  dollars  a  year  forever  ? 

754.  How  long,  to  endow  each  of  the  three  colleges  in  the 
state  with  a  fund  for  the  support  of  indigent  students,  which 
will  forever  maintain  200  -uch  students  at  each  college,  at 
200  dollars  each  per  annum  ? 

755.  How  long,  that  the  saving  may  furnish  each  of  the 
towns  in  the  state,  (being  then  452,)  with  a  public  library  of 
1000  volume*,  at  Sl*5'  a  volume  ? 

756.  If  the  Western  Education  Society  allow  their  bene- 
fic-aries  70  dollars  a  year  each,  and  the  people  of  the  state  of 
New- York  should  abstain  from  ardent  spirits  only  half  the 


Questions.  203 

time,  and  pay  the  amount  into  the  treasury  of  that  Society, 
how  many  young  men  would  it  enable  them  to  assist  annually 
M  that  rate  ? 

'  757.  What  is  the  share  of  the  state  of  Vermont,  in  the 
annual  expense  of  ardent  spirits  -' 

758.  How  long  must  the  people  of  that  state  abstain 
from  ardent  spirits,  that  the  saving  may  endow  each  of  their 
two  colleges  with  a  fund,   which,  at  6  per  cent,  shall  yield 

'^5000  dollars  a  year  forever  ? 

759.  How  long,  to  endow  an  academy  in  each  of  the  13 
counties  of  that  state,  with  a  fund,  which,  at  6  per  cent,  shall 
yield  600  dollars  a  year  forever? 

760.  \V  hat  is  the  share  of  the  state  of  Connecticut  ? 

761.  How  long  must  the  people  of  that  state  abstain  from 
ardent  spirits,  that  the  saving  may  support  a  minister  of  the 
gospel  in  each  of  the  2i9  parishes  of  tnat  state,  at  600  dolls, 
each  per  annum  ? 

762.  How  long,  to  support  a  charity  school  in  each  of  the 
8  countiefi  of  that  state,  allowing  0  scholars  to  each  school, 
at  ^00  dollars  each,  and  the  instructor  700  dollars  a  year  ? 

763.  How  long,  that  the  saving  may  furnish  each  parish 
''  with  a  public  library  of  K  00  volunies,  at  !S  -•'  0  a  volume  ? 

764.  How  hmg,  to  furnish  Yale  College  with  a  fund, 
which,  at  6  per  cent,  will  support  200  charity  students,  at 
£20'  each  per  annum,  forever  ? 

7C5.  How  long,  to  furnish  said  College  with  a  fund  for  the 
support  of  additional  professors,  which  shall  yieiil  oCGO  dulls. 
a  year  forever  ? 

766.  How  long,  to  build  a  place  of  worship  in  each  parish 
in  the*  state,  at  6000  dollars  each  ? 

767.  If  the  grand  list  of  that  state  is  5959756  dollars  ; 
how  much  on  the  dollar  is  the  annual  expense  of  ardent 
spirits  ? 

■  768.  How  many  miles  would  the  share  of  Connecticut 
»reach,  at  the  rate  stated  m  No.  749  -' 

769.  What  is  the  share  of  the  state  of  Massachusetts  ? 

770.  How  many  indigent  students  would  it  enable  the 
American  Education  Society  to  assist,  at  125  dollars  each  ? 

771.  ^^  hat  is  the  share  of  New-England  ? 

77^:.  How  many  missionaries  would  it  support,  at  500 
dollars  each  ? 

?7.S.  How  many  free  schools  would  it  support  in  India,  at 
24  dollars  each'? 


204  Questwns, 

774.  How  many  heathen  children  would  it  keep  at  Christ 
tian  schools  continually,  at  50  for  each  .' 

775.  To  how  many  heathen  children  would  the  saving  of . 
20  years  give  a  Christian  education,  allowing  each  child  to 
remain  at  school  4  years  ?  1 

776.  How  many  mini*^ters  of  the  gospel  would  the  saving 
for  S4  years  educate,  allowing  them  to  spend  8  years  in  their 
preparation,  at  200  dollais  each  per  annum  ? 

77^.  If  all  the  people  in  the  state  of  New-York  should 
contribute  one  cent  a  week,  each,  to  the  Western  Education 
Society,  how  many  young  men  would  it  enable  them  to  as- 
sist, at  the  rate  mentioned  in  No.  75  6  ? 

778.  If  there  should  be  a  charity  school  established  in 
each  of  the  8  counties  of  Connecticut,  and  the  instructor' of 
each  should  receive  7<.0  .dollars  a  year,  and  there  should  be 
SO  charity  scholars  at  each  school,  at  an  expense  of  i  50  dol- 
lars each  a  year,  how  mu^h  would  be  the  expense  per  week 
to  each  pe»  sou  in  the  state  •' 

77'J.  II  evrry  per<^on  in  Connecticut  should  contribute  one 
€ent  a  week,  how  many  rp/issionaries  v/ould  it  support,  at 
500  dollars  each  per  annuo;  ? 

780.  if  every  pprson  in  Massachusetts  should  contribilc 
one  cent  a  v\e»jk,  how  many  orphans  would  it  support  m  In- 
dia, at  1  ^  dollars  each  .^ 

781.  [f  every  person  in  Vermont  should  contribute  one 
cent  a  week,  how  many  bibles  w  )u'd  it  pay  for  anriuaily,  at 
60  cents  each  / 

78^.  If  every  person  in  New-Han}rM?re  f!r>old  cojitribute 
one  cent  a  week,  how  ma  >y  tru'ts  ol  ; 'i  pages  eacii,  would 
it  pay  for  annually,  at  1  nri*  a  |»a«,t  T 

78  ^.  if  every  person  in  the  Un;red  Siates  should  contri- 
bute one  cent  a  week,  for  doing  gtiod,  how  much  would  be 
thus  raised  annually  ? 

7H4.  How  many  missionaries  would  the  half  of  it  support, 
at  500  dollars  a  year  each  ? 

78  .  How  many  young  men  would  the  quarter  of  it  assist 
in  preparing  tor  the  ministry,  at  *25  dollars  a  year  each  ? 

786.  How  many  bibles  would'the  eighth  of  it  pay  for,  at 
60  cents  each  ? 

787.  How  many  tracts,  of  12  pages  each,  would  the  rest 
pay  for,  at  I  mill  a  page  ^ 

788.  According  l<i  the  reports  made  to  the  General  \s- 
seuibly  of  the  Presoyleuan  Church  in  the  United  States,  in 


Questions,  ^0^ 

1819,  the  number  of  Presbyteries  under  their  care,  was  as 
follows  :  Synod  of  Geneva,  6  ;  Albany,  6  ;  New- York  and 
New-Jersey,  6;  Philadelphia,  6;  Pittsburgh,  6;  Vu'ginia, 
4 ;  Kentucky,  4  ;  Ohio,  4  ;  Tennessee,  5  ;  North-Carolina, 
S  ;  South-Carolina  and  Georgia,  3  :  how  many  in  all  ? 

789.  At  the  same  time,  the  number  of  ordained  ministers 
in  the  Synod  of  Geneva,  and  the  number  of  congregations 
under  their  care,  was  as  follows  : 

Presbytery  of  Jlfhi,  Coii^.  Presbytery  of  JYlln,  Cong, 

Niai^'ara,  10  32  Geneva,  17  '2^i' 

Ontario,  20  23     I     Cayuga,  19  28 

Bath,  6  11     '    Onondaga,  21  29 

How  many  ministers,  and  how  many  congregations  ? 

79  .  In  the  Synod  of  Albany,  as  follows  \ 


Albany,  16  22 

Coui'iihia,  13  23 

Oneitla,  25 


Londorulerry,  18  13 

Gtiamphun,  10  13 

Lawrence,  12  4 


Jcrse\ ,  28  29 

New-Brunswick,      15  IG 

Newton,  14  25 


How  many  ministers,  and  how  many  congregations  ? 
7   1.  In  the  Synod  of  N.  York  &  N.  Jersey,  as  follows  : 

Long- Island,  16  16  ' 

Hua'sou,  22  39 

Nevv   York,  13  22 

How  many  miribters,  and  how  many  congregations  ? 

792.  In  th^.  feyaod  of  Philadelphia,  as  follows  : 

Philadelphia,  26  37     ,     Cadisle,  29  36 

New-Gastle,  27  51  Huntingdon^  12  29 

Baltufiore,  16  12     i     N^Tthumbtrland,      7  I'j 

How  many  ministers,  and  how  mai  y  congregations  ? 

793.  In  the  Synod  of  Pittsburgh,  as  follows  : 

Redstone,  19  23  Hartf.M-d,  9  25 

Ohio,  28  48     I     Gv-and  'liver,  G  16 

Erie,  12  45     I      Portage,  7  20 

How  many  ministers,  and  how  many  con2;regations  ? 

794.  In  the  Synod  of  Virginia,  asjollows: 

lla..uver,  15  26     j      Winchester,     -        13  15 

Lexington,  16  30     |     Abington,  7  10 

How  many  ministers,  and  how  many  congregations  ? 

795.  In  the  Synod  of  Kentucky,  as  foUq^ws  : 

Transylvania,  9  ^^1     Muhlenberg,  5  22 

West  Lex.ngton,      12  27     j     Louisvihe,  11  31 

How  many  ministers,  and  how  many  congregations  ? 

796.  in  the  Synod  of  Ohio,  as  follows  : 

Wa!»hington,  9  26     I     Miami,  14  S6 

Lancaster,  15  34     J     Richland,  7  2'J 

How  many  ministers,  and  how  many  congregations  ? 

797.  In  the  Synod  of  Tennessee,  as  follows : 

Union,  9  16     I     *Miisissipi>i,  5  S 

West  Tennessee,       6  16  Missouri,  4 

*Shiloh,  7  10    I 

How  many  ministers,  and  how  many  congregation'^^ 
•S  •  Estimates, 


206  Questions, 

798.  In  the  Synod  of  North-Carolina,  as  follows  : 


Presbytery  of  JMiiu  Co7ijf, 

Concordi  16  68 


Prestytery  of  JWn»  Cong, 

Orange,  10  22 

Fayetteville,  il  32 

How  many  ministers,  and  how  many  congregations  ? 

799.  In  the  Synod  of  S.  Carolina  &  Georgia,  as  follows  : 

Harmony,  19  «8     I     Hopewell,  6  15 

S.  Carolina,  15  30     j 

How  many  ministers,  and  how  n^any  congregations  ? 

800.  Take  the  whole  number  of  ministers,  and  the  whole 
number  of  congregations  in  the  preceding  eleven  questions, 
and  add  104  licensed  preachers,  not  ordained  ;  and  tell  the 
whole  number  of  authorised  preachers,  and  of  congregations, 
in  connection  with  the  General  Assembly,  in  the  year  1819. 

8' a.  If  the  whole  number  of  communicants  in  all  the 
churches  in  the  United  States,  should  be  equally  divided 
into  8  classes,  how  many  would  be  in  each  class  ? 

802.  If  the  first  class,  comprising  the  most  wealthy  in  our 
large  towns,  should  contribute  each  50  dolls,  a  year  for  cha- 
ritable purposes,  what  would  be  the  annual  amount? 

803.  If  the  second  class,  comprising  the  most  wealthy  in 
the  country  villages,  give  each  SO  dollars,  what  would  be 
the  amount  ? 

804.  If  the  third  class,  comprising  those  less  wealthy,  give 
each  lO  dollars,  what  would  be  the  amount? 

805.  If  the  fourth  class,  comprising  farmers,  mechanics, 
&c.  give  each  5  dollars,  what  would  be  the  amount  ? 

806.  If  the  fifth  class,  comprising  young  men,  give  each  3 
dollars,  what  would  be  the  amount  ? 

807.  If  the  sixth  class,  comprising  young  w^omen,  give 
each  2  dollars,  what  would  be  the  amount? 

808.  If  the  seventh  class,  comprising  the  poorer  sort,  who 
enjoy  health,  and  can  labor,  give  each  one  dollar,  what  would 
be  the  amount? 

809.  If  the  eighth  class,  comprising  the  ag;ed,  the  infirm 
the  feeble,  &c.  give  each  25  cents,  what  would  be  the  amountr 

810.  What  would  be  the  whole  amount  annually  contri- 
buted by  these  eight  classes  ? 

811.  if  they  should  all  continue  their  contributions  for  10 
years,  W'hat  would  be  the  amount  ? 

812.  If  this  was  made  a  permanent  fund,  what  annual  in- 
terest would  it  yield,  at  6  per  cent  ? 

813.  If  half  that  interest  should  be  applied  to  the  educa- 
tion of  young  men  for  the  gospel  ministi  y,  how  many  v/ould 
it  assist,  at  g  1 25  each  ? 


Questions,  207 

"^4.  If  the  other  half  should  be  applied  to  the  support  of 
issioiiaries,  how  many  would  it  maintain,  at  8500  each  ? 

815.  In  1817,  the  receipts  of  the  British  National  School 
Society  for  educating  the  poor,  on  the  Lancasterian  system, 
had  been,  for  6  years,  SS25QI.  sterling ;  how  much  is  that  in 
federal  money  ? 

816.  The  number  of  children  in  their  schools  in  that  year, 
was  155000;  if  one  fourth  of  the  above  sum  was  expended 
in  tbeir  education  in  that  year,  how  much  is  that  for  each 
child? 

817.  It  is  estimated,  that  there  are  64  millions  of  children 
in  the  world  at  a  time,  who  ought  to  be  at  school  ;  and  if 
each  child  can  be  schooled,  on  the  British  system,  at  30 
cents  a  year,  what  would  it  cost  for  the  whole  ? 

818.  What,  for  22  years  ? 

819.  If  5  persons  are  allowed  to  a  family,  how  many  fa- 
milies are  there  in  the  whole  world  ?     (See  No.  i3.) 

820.  If  5  millions  of  families  are  already  supplied  with 
bibles,  and  a  bible  costs  60  cents,  how  much  would  it  cost  to 
supply  the  rest  of  the  world  with  one  to  a  family  ? 

821.  If  a  preparation  of  8  years,  at  an  expense  of  203 
dollars  a  year,  is  necessary'  for  a  missionary  ;  what  is  the 
whote  amount  ? 

822.  If  one  missionary  is  v/anted  for  every  3000  of  those 
who  are  not  nominal  Christians,  what  would  it  cost  to  edu- 
cate the  requisite  number?     (See  Nos.  13  and  14.) 

823.  If  it  should  cost  as  much  to  convey  each  one  to  his 
station,  as  the  passage  from  i^merica  to  India,  which  is  250 
dollars  ;  v/liat  would  it  cost  to  convey  the  whole  number  to 
their  stations  r 

824.  What  would  it  cost  to  maintain  the  wli(4^r"22  years 
at  .500  dollars  a  year  each  ?  ' 

825.  If  40000  additional  ministers  are  wanted  to  supply 
the  destitute  in  Christian  countries,  what  would  it  cost  to 
educate  them,  and  support  them  2'^  years,  at  the  above  rates? 

826.  Wliat  would  it  cost  to  scliool  all  tlie  cluklren  in  the 
world,  for  22  years  ;  to  supply  the  whole  world  with  bibles  ; 
and  to  educate  and  support  for  22  years,  a  supply  of  ministers 

iid  missionaries  for  all  the  destitute  in  the  world  ? 

827.  During  the  war  consequent  upon  the  French  revoiu 
tion,  from  1793  to  1815,  a  period  of  22  years,  the  war  expense 
of  Great  Britain  is  calculated  to  have  been  3200  millions  of 
dolls.;  France,   3130  millions ;  Austria,   1000  millions;  U. 


£08  %iestwns. 

States,  (3  years,)  120  millions  ;  other  powers  of  Europe,  es- 
timated at  4550  millions  :  what  was  the  whole  expense  of 
that  war  to  the  nominally  Christian  vvorld  i* 

828.  If  they  had  been  willing  to  expend  one  third  as  much 
to  inform,  moralize,  and  chri^^tianize  the  world,  how  much 
more  than  sufficient  would  that  sum  have  been,  to  accomplish 
all  these  objects,  during  the  same  period,  according  to  the 
preceding  statements  ? 

829.  If,  instead  of  an  appeal  to  arms,  the  nations  had  esta- 
blished a  general  congress  of  all  the  Christian  powers,  for  the 
settlement  of  all  difficulties  between  nations  ;  and  the  above 
surplus  had  defrayed  its  expenses  for  the  same  period,  how 
much  would  it  have  been  for  each  year  ? 

830.  In  the  preceding  estimate  of  the  expense  of  a  single 
war,  it  is  probable  that  nothing  is  included  but  the  sums  ac- 
tually paid  out  by  the  respective  governments  ;  and  that  the 
loss  of  productive  labor,  the  loss  of  lives,  and  the  destruction  of 
private  property,  are  omitted.  The  military  w^r  establish- 
ment of  Europe  is  stated  at  3908000  men.  Suppose  only  2 
millions  were  actually  under  arms  during  those  22  years,  and 
9.  millions  more  were  employed  m  preparing  and  conveying 
arms  and  stores  ;  what  is  the  loss  of  productive  labor,  reck- 
oning these  men  to  have  been  able  to  earn  only  30  cents  a 
day,  at  some  useful  employment,  excluxling  the  Sabbaths, 
ana  including  the  additional  days  for  leap  years  ? 

831.  If  the  number  of  lives  lost  in  the  first  seven  years  of 
that  war,  was  in  the  same  proportion  as  in  the  last  15,  (see 
No.  696,)  how  many  lives  were  lost  in  all  ? 

832.  If  the  pecuniary  loss  to  the  public,  from  the  death  of 
m  able  bodied  man,  is  S1500  ;  what  is  the  amount  of  this 
item  of  loss  by  that  war  > 

833.  What  is  the  total  amount  of  the  loss  of  labor  and  the 
loss  of  lives? 

834.  Among  the  numerous  idolatrous  festivals  of  the  Hin- 
doos, it  is  computed,  that  the  annual  expense  of  one  to  the 
inhabitants  of  Calcutta,  is  500000/.  steiling  ;  what  would  it 
be  to  the  whole  of  Hindoostan,  at  the  same  rate  ■?     (See  No* 

60;,  mr.) 

8  5.  How  n^any  missionaries  would  it  support,  at  II SL 
each  ? 

836.  If  the  whole  population  of  Hindoostan  were  divided 
Hmong  them,  how  many  souls  would  it  be  to  each  ? 


#        ^uestianh  2bd 

837.  What  was  the  solid  content,  in  feet,  ot  Noah's  ark, 
^being  300  cubits  long,  50  wide,  and  30  high  ? 
' '  838.  It  is  stated  by  the  learned,  that  there  are  about  150 
kinds  of  quadrupeds,  200  kinds  of  birds,  and  40  kinds  of 
reptiles,  that  must  have  been  preserved  in  the  ark.  Suppose 
there  were  £00  kinds  of  quadrupeds,  of  which  20  were  clean; 
300  kinds  of  birds,  of  which  20  were  clean  ;  and  50  kinds  of 
reptiles  ;  and  that  7  of  every  kind  of  clean  animals,  and  2  of 
every  kind  of  unclean  ones,  and  8  persons,  were  preserved  ; 
how  many  living  creatures  would  there  be  ? 

839.  Suppose  the  quadrupeds  to  average  the  size  of  a  two 
year  old  steer,  and  to  require  stalls  that  should  give  each  8 
feet  in  length,  5  in  width,  and  6  in  height ;  the  birds  to  ave- 
rage the  size  of  a  hen,  and  to  require  lofts  that  should  give 
each  2  cubic  feet  of  room ;  and  the  reptiles  to  have  i  cubic 
foot  each  :  how  many  cubic  feet  would  all  these  animals 
occupy  ? 

840.  Allow  Noah  and  his  family,  1  kitchen,  20  ft.  long,  20 
wide,  and  10  high  ;  i  parlor,  and  1  store- room  for  provisions* 
of  the  same  dimensions  ;  and  4  lodging  rooms,  each  10  ft.  long, 
10  wide,&  10  high  ;  how  many  cubic  feet  would  they  occupy  ? 

841.  If  one  third  of  the  unclean  beasts  and  one  fourth  of 
the  unclean  birds  were  carnivorous,  and  2  sheep  a  day  were 
allowed  for  6  beasts  and  7  birds,  and  each  sheep  required  62 

'cubic  feet  of  room  ;  how  many  cubic  feet  would  be  occupied 
by  the  sheep  necessary  in  a  year  ? 

842.  If  each  of  these  sheep  was  allowed  a  quart  of  grain  a 
day,  and  an  equal  quantity  of  water,  till  it  was  killed,  how 
many  cubic  feet  would  be  necessary  to  store  that  grain  and 
water  ? 

843.  If  each  of  the  rest  of  the  beasts  was  allowed  a  peck 
of  grain,  and  an  equal  quantity  of  water  per  day  ;  and  each 
of  the  rest  of  the  birds,  one  eighth  of  a  quart  of  grain,  and  an 
equal  quantity  of  water  per  day  ;  how  many  cubic  feet  would 
be  necessary  to  store  that  grain  and  water  ? 

844.  If  each  of  the  sheep  kept  for  food  was  allowed  :J  of  a 
cubic  foot  of  pressed  hay  per  day,  till  it  was  killed ;  and 
each  of  the  other  beasts  not  carnivorous,  was  allowed  2  cubic 
feet  per  day,  through  the  year  ;  how  many  solid  feet  would 
be  necessary  to  store  that  hay  ? 

^4i.  Allow  Noah's  family  to  use  50  gas  ale  measuje, 
water  per  day  j  how  many  cubip  feet  would  be  necessary  to 
store  it  ? 

1S2 


210  i^uestions.         ^^ 

846.  Allow  Noah  3  rooms,  each  20  ft.  long,  20  wide,  and 
10  high,  to  store  farming  utensils,  and  other  necessary  arti 
cies  ;  how  many  cubic  feet  would  they  occupy  ? 

84/.  Allow  JO  ft.  wide,  at  one  end  of  the  ark,  the  whole 
breadth  and  height,  for  stair  cases;  and  10  feet  wide  diroisgl 
the  whole  remaining  length,  and  the  whole  height,  for  pas- 
sages ;  and  how  many  cubic  feet  would  be  occupied  by  these 

848.  Add  the  preceding  nine  items  together,  and  how 
many  cubic  feet  wouJd  be  left  for  the  thickness  of  the  walls, 
floors,  partitions,  and  other  purposes  ? 

849.  The  contributions  to  the  funds  of  the  Americar 
Board  for  Foreign  Missions,  for  the  year  ending  Aug.  31, 
1821,  were  as  follows  :  From  Massachusetts,  19820-66  dolls.; 
Connecticut,  7874-08  ;  New  York,  6424-96  ;  Vermont,  1912 
•96:  N.  Hampshire,  1699*40;  Maine,  1429-76;  N.Jersey, 
1384-74;  Pennsylvania,  1312-99  ;  Georgia,  1052-36;  Ohio, 
506-10;  S.  Carolina,  495-06  ;  Virginia,  400-62;  Maryland, 
393-50;  Kentucky,  369-31;  N.  Carolina,  251-30;  Rhodes 
Island,  65-56  ;  Tennessee,  63-00  ;  Delaware,  36^00  ;  Mi- 
chigan, 26-75  ;  Columbia  Dist.  15-00;  Indiana,  8-43  ;  Choc- 
taw Nation,  74-25;  Cherokee  Nation,  31-00;  Switzerland, 
212  00;  England,  40-00;  S.  America,  3  C^O ;  West  Indies, 
3-00  ;  places  unknown,  491 '89  :  how  much  in  all  ? 

850.  How  much  less  than  half  the  whole,  was  contributed 
by  Massachusetts  ? 

85 . .  How  much  more  than  half  the  whole,  by  Massachu- 
setts and  Connecticut  ? 

852.  How  much  is  the  amount  contributed  by  Massachu- 
setts, for  each  person  in  the  state  ?     (See  No.  404.) 

853.  How  much  is  the  amount  contributed  by  Connecticut, 
for  each  person  in  the  state  I 

85 1.  If  New  York  had  contributed  in  the  same  proportion 
as  Massachusetts,  what  sum  would  have  been  raised  in  that 
s^ate? 

8.^5-  If  all  the  states  had  contributed  in  the  same  propor* 
tion  as  Massachusetts,  what  sum  would  have  been  raised  in 
the  whole  r 

856.  If  the  extra  expense  of  the  thanksgiving  dinner,  to 
the  inhabitants  of  New-England,  reckoning  time  and  money, 
is  30  cents  each  person;  andif  they  should  satisfy  themselves 
witb.  a  common  dinner  on  that  day,  and  testify  their  gratitude 
to  God  by  devoting  that  amount  to  his  service  ;  how  much 
•would  be  thus  raised  annuallv? 


m 


Questions.  211 


857.  If  all  the  people  of  the  United  States  should  content 
themselves  with  a  dinner,  once  a  week,  that  should  cost  one 
cent  less  than  ordinary,  for  each  person,  and  should  pay  that 
amount  into  the  treasury  of  the  Lord  ;  how  niuch  would  be 
tlius  raised  annually  ? 

85b.  Take  the  receipts  of  the  principal  charitable  societies 
in  En<:;land,  as  stated  in  No,  54,  and  suppose  all  the  other 
charitable  societies  in  Europe  to  do  as  much  as  the  British 
and  Foreign  Bible  Society,  and  what  is  the  annual  amount, 
in  federal  money,  contributed  in  Europe  for  spreading  the 
gospel ? 

859.  During  the  year  ending  in  1821,  the  American  Bible 
Society  received  49578  dolls. ;  the  American  Board  for  Fo- 
reign Missions,  463^8  dolls. ;  the  Baptist  Board  for  Fore:gn 
Missions,  18000  dolls. ;  the  United  Foreign  Mission  Society, 
15263  dolls. ;  the  American  Education  Society,  13109  dolls.; 
and  30  smaller  societies,  about  60388  dolU.  Add  60000 
dolls,  more,  for  all  other  societies  ;  and  what  is  the  annual 
amount  contributed  in  the  United  States  for  spreading  the 
gospel ? 

860.  How  long  would  the  people  of  the  United  States 
need  to  abstain  from  ardent  spiiits,  to  save  that  amount, 
supposing  the  same  quantity  to  be  consumed  now,  that  was 
in  1810?     (See  No.  734.) 

861.  How  long,  to  save  as  much  as  is  contributed  in  both 
Europe  and  America  ? 

'  862.  Nearly  all  the  societies  in  the  world  for  the  spread 
of  the  gospel,  have  been  formed  within  30  years,  and  most  of 
them  within  half  that  period.  It  is  calculated,  partly  from 
documents,  and  partly  from  estimates,  that  the  following 
sums  have  been  raised  for  that  purpose,  by  the  principal  so- 
cieties in  England,  during  that  period,  to  wit :  By  the  British 
and  Foreign  Bible  Society,  403^658  dolls.  ;  by  13  others, 
7704222.  Suppose  all  the  other  societies  in  Europe  to  have 
done  as  much  as  the  B.  &  F.  Bible  Society  ;  and  what  is  the 
whole  amount  v\  lich  has  been  raised  in  Europe  for  the  spread 
of  tlie  gospel,  within  Hie  last  30  years  ? 

863.  What  is  the  average  amount  per  year  ? 

864.  it  is  thought,  that  if  we  allow  to  the  American  So- 
cieties an  average  income  for  20  years,  which  shall  bear  the 
same  proportion  to  their  present  income,  that  the  above 
stated  average  income  of  the  European  Societies  bears  to 
their  present  income,  it  will  equal  th«  whole  amount  of  what 


21^  Questions. 

has  been  done  in  the  United  States  for  the  spread  of  the 
gospel  in  30  years.     If  so,  what  is  the  amount? 

86^.  What  is  the  whole  amount  of  what  has  been  done  for 
that  object,  in  Europe  and  America,  during  30  years  ? 

866.  How  long  would  the  people  of  the  United  States  need 
to  do  without  ardent  spirits,  to  save  that  amount  ?  (See  So. 
734.) 

867.  It  is  computed,  that  200  bushels  of  potatoes,  or  some- 
thing equivalent,  can  be  raised  in  a  missionary  field,  by  labor 
equal  to  36  days'  work.  If  this  is  so,  and  one  person  in  4 
of  the  whole  population  of  the  United  States  should  labor  S 
days  every  season  for  that  jiurpose,  and  the  potatoes  should 
sell  for  b  sixteenths  of  a  dollar  per  bushel ;  what  amount 
would  be  thus  raised  annually  ? 

868.  If  one  person  in  ten  of  the  whole  population  of  the 
United  States,  sleeps  an  hour  later  every  morning  than  is 
necessary,  and  his  labor  is  worth  8  cents  an  hour ;  what  an- 
nual amount  is  thus  lost,  which  might  be  saved  for  doing 
good,  reckoning  313  working  days  '? 

869.  If  a  youth  can  read  an  octavo  page  in  2  minutes,  and 
is  in  the  habit  of  spending  3  hours  each  evening  in  idleness, 
from  the  first  of  November  to  the  first  of  March  in  each  year ; 
how  many  volumes,  of  300  pages  each,  could  he  read,  in  his 
winter  evenings  for  10  years  ? 

870.  In  the  year  1821,  it  was  supposed  there  were  .2500 
dram  shops  kept  in  the  city  of  New  York.  If  the  rent  of 
ea*  h  of  these  is  70  doli^.,  and  the  labor  of  one  man  to  attend 
each,  is  worth  200  dolls  ;  what  is  the  amount  of  these  two 
items  of  the  expense  of  these  public  nuisances  ? 

871  If  each  of  these  shops  sells  liquor  to  the  amount  of 
only  2  dolis.  per  day,  what  is  the  annual  amount  ? 

872.  In  1820,  there  was  stated  to  be  SvOO  paupers  in  the 
city  of  New- York.  If  the  expense  of  supporting  them  ave- 
rages 1  doll,  per  week,  what  is  the  annual  amount  ? 

873.  It  is  believed,  that  three  fourths  of  the  expense  of 
supporting  paupers  is  occasioned,  directly  or  indirectly,  by 
intemperance.  Take  |  of  the  last  amount,  and  add  to  the 
two  preceding,  and  tell  the  total- 

874.  During  the  same  year,  the  amount  expended  for  pub- 
lic and  private  schools,  was  $14759*41  ;  and  there  were  63 
ministers  of  the  gospel  employed  in  the  city.  If  these  had  a 
salary  of  $1500  each,  what  amount  was  expended  for  reli- 
gious and  literary  instruction  ? 


^ueshon^,  213 

By  a  report  made  to  the  legislature  of  Massachusetts 
In  182  I,  It  appeared,  that  the  number  of  paupers  in  that  state 
was  about  one  sixty-feixth  of  the  whole  population  ;  and  that 
the  average  expense  of  supporting  them  was  one  doHar  per 
week.  If  it  is  the  same  in  all  the  other  states,  and  three 
fourths  of  the  expense  i»  occasioned  by  intemperance,  what 
amount  would  be  annually  saved  in  this  way,  by  the  disuse 
of  ardent  spirits  ? 

71  876.  In  1812,  Mr.  H.  Campbell  estimated  the  poor  rates  in 
England  and  Wales,  at  1 64 j26o 6/.  sterling,  and  the  number 
of  paupers  at  20794S2  ;  what  is  that,  in  federal  money,  for 
each  pauper  ? 

877.  In  i8£0,  it  was  stated  that  there  were  14000  paupers 
in  Liverpool,  supported  by  parish  rates  paid  by  20000  indivi- 
duals ;  if  each  pauper  cost  as  stated  in  the  last  question, 
what  had  these  individuals  to  pay  on  an  average  ? 

878.  The  Connecticut  Missionary  Society  paid  for  mis- 
sionary services,  in  1816,  85466*38,  of  which  $45.::'6l  were 
for  services  rendered  in  Kentucky,  jg  1328*38  in  New- York, 
$6lo-85  in  Missouri,  $o-ll  in  the  sod thern  part  of  Ohio, 
^394-93  in  Pennsylvania,  $367  in  Tennessee,  S294-28  in 
Vermont,  $150  in  Indiana,  and  the  rest  in  New  Connecti- 
cut ;  hbw  much  was  the  last  ? 

879.  In  1730,  the  number  of  graduates  at  Yale  College  for 
29  years,  had  been  235,  of  which  118  became  ministers  of  the 
gospel ;  how  many  ministers  does  that  average  yearly  ? 

880.  How  many  did  not  become  ministers,  to  one  that  did? 

881.  In  the  next  20  years,  to  1750,  there  were  385  gradu- 
ates, of  which  162  became  ministers;  how  many  ministers 
yearly? 

I     882.  How  many  did  not  become  ministers  to  one  that  did  ? 

v^  883.  In  the  next  20  years,  to  1770,  there  were  648  gradu- 
ates, of  which  201  became  ministers  ;  how  many  ministers 
yearly  ? 

884.  How  many  did  not  become  ministers,  to  one  that  did? 

885.  In  the  next  20  years,  to  1790,  there  were  723  gradu- 
ates, of  which  177  becatne  ministers;  how  many  ministers 
yearly  ? 

886.  How  many  did  not  become  ministers,  to  one  that  did? 

887.  In  the  next  20  years,  to  1810,  there  were  790  gradu- 
ates, of  which  160  became  ministers  ;  how  many  minister^ 
yearly  ? 

888.  How  many  did  not  become  ministers,  to  one  that  did? 

889.  In  1767,  the  graduates  at  Princeton  College  for  %0 


214  Questions, 

years,  had  been  S§1,  of  which  130  became  ministers  ;  how 
many  ministers  yearly  ? 

890.  How  many  did  not  become  ministers,  to  one  that  did  ^ 

891.  In  1807,  the  graduates  for  20  years' had  been  525,  oi 
which  50  became  ministers  ;  how  many  ministers  yearly  ? 

892.  How  many  did  not  become  ministers,  to  one  that  did  ? 

893.  Harvard  University,  from  1719  to  1741,  furnished  an 
average  of  13  ministers  annually  ;  how  many  in  the  whole  in 
that  time  ? 

894.  From  1800  to  1810,  only  6  annually;  how  many  in 
that  time  ? 

895.  Dartmouth  College,  from  1780  to  1800,  furnished  an 
average  of  8  ministers  annually  ;  how  many  in  that  time  ? 

896.  From  18G0  to  1810,  only  5  annually;  how  many  in 
that  time  ? 

897.  The  proportion  of  graduates  at  the  principal  colleges, 
who  entered  the  ministry,  from  1800  to  iSi),  was  one  sixth  ; 
and  Harvard,  Yale,  Union,  and  Princeton,  together,  sent  out 
about  200  graduates  a  year ;  how  many  ministers  did  they 
furnish  in  that  time? 

898.  U  all  the  other  colleges  in  the  United  States  fur- 
nished as  many  more  ;  what  is  the  whole  in  those  1 0  years  ? 

899.  If  the  number  of  educated  ministers  in  the  United 
States,  in  1820,  was  2390  :  and  there  should  be  500  gradu- 
ates annually  at  the  colleges,  of  which  one  sixth  should  be- 
come ministers,  and  half  as  many  more  should  obtain  a  sufR- 
cient  education  without  going  to  college,  and  none  should 
die  ;  how  many  would  there  be  in  the  ^ear  1 843  ? 

900.  But  if  the  life  of  a  minister  averages  25  years  after 
entering  the  ministry,  only  2  twenty-fifths  of  the  old  ones, 
and  about  1 6|  twenty -fiftlis  of  the  new  ones,  will  then  be 
alive  ;  how  many  will  that  be  ? 

901.  If  the  population  of  the  United  States  should  double 
in  *t3  years,  and  one  minister  is  necessary  for  every  bOO 
souls,  how  many  would  then  be  destitute  of  an  educated  mi- 
nistry ? 

902.  In  the  next  23  years,  let  801  be  graduated  antmally, 
and  one  fifth  of  them  enter  the  ministry,  and  half  as  man3r 
more  without  going  to  college  ;  how  many  ministers  would 
there  be  then,  after  deducting  the  deaths  as  above  ? 

903.  If  t'iie  population  doubles  again,  how  many  will  be 
destitute  in  the  year  1 866  ? 

904.  In  the  next  23  years,  let  lOOO  be  graduated  annually, 
and  one  fifth  of  them  enter  the  ministry,  and  half  as  many 


Questions,  215 

wore  without  going  to  college  ;  how  many  ministers  would 
/there  be  then,  after  deducting  the  deaths  as  above  ? 

905.  If  the  population  doubles  a^ain,  how  many  will  be 
destitute  in  the  year  1889  ? 

906.  If  the  whole  number  of  Protestants  is  60  millions, 
and  only  50000  missionaries  are  wanted  to  send  to  the  hea- 
then, and  if  the  number  of  Protestants  in  the  United  States 
is  93750^0,  how  many  ought  they  to  furnish  ? 

907.  The  donations  to  the  Massachusetts  Missionary  So- 
ciety, for  18S0,  were  $2371v^T,  of  which  g  1341-33  was  for 
the  permanent  turd  ;  how  much,  in  N.  England  currency, 
was  for  current  expenses  ? 

908.  The  expenditures  of  the  Massachusetts  Christian 
Knowledge  Society,  for  1820,  were  $1456*3:>;  in  18  l6  they 
were  $2622-33  ;  how  many  guilders  was  the  decrease? 

909.  The  receipts  of  the  Lf>ndon  Jew^'  Society,  in  1817, 
were  1-10091  sterling  ;  how  many  rials  of  plate  is  that  ? 

910.  The  VernunJ  Bible  Soc.  received,  in  i8l4,$l462'13; 
how  many  English  guineas  is  that  ? 

911.  The  Connecticut  Education  Society,  in  1817,  receiv- 
ed ^'1 370*43  ;  how  many  millrees  is  that  ? 

912.  The  Female  Education  Society  of  New-Haven,  re- 
ceived $351*18  ;  how  many  rupees  is  that  ? 

913.  The  Connecticut  Domestic  Missionary  Society  re- 
ceived 81263-63;  how  many  rubles  is  that? 

914..  The  permanent  fund  of  the  Connecticut  Missionary 
Society,  was  S3 1583  65  ;  how  many  Hebrew  shekels  of  silver 
is  the  annual  interest  at  6  per  cent  ? 

915.  I'he  Philadelphia  Education  Society,  in  1890,  re- 
ceived $2039-8^ ;  how  many  Greek  oboli  is  that  ? 

91G.  In  1819,  the  Female  Missionary  Society  of  the  Wes- 
tern District  of  N- York,  received  $1352-38:  how  many 
Roman  sestertii  is  that  ? 

917.  In  1820,  the  General  Assembly  of  the  Presbyterian 
Church  appropriated  for  their  theological  Seminary,  $47i2 
•055,  of  which  were  expended  as  follows  :  Salaries  and 
house  rent  oi  2  professors,  $4000  ;  half  year's  salary  of  as- 
sistant teacher  of  languages,  $200 ;  printing,  stationary,  &c. 
$40-07i  ;  travelling  expenses  of  one  director,  $21 ;  treasu- 
rer's commissions  on  the  above,  $42-61  :  how  many  drachmae 
of  silver  in  what  was  unexpended  ? 

918.  In  1818,  the  Western  Education  Society  received 
g^;028'67,  and  expended  $1531-53  ;  how  many  Hebrew  she- 
kels of  gold  in  the  difiereuce  ? 


^16  Questions. 

#'9.  In  1819,  the  N.  York  Religiaus  Tract  Society  re- 
ceived B849*95  ;  how  many  Greek  staters  of  silver  is  that  ?  ^ 

920.  In  1 8 19,  the  collections  for  the  Theological  Seminary 
of  the  Dutch  Reformed  Church,  were  $3730'04  ;  how  many 
Roman  denarii  is  that  ? 

921.  The  funds  of  their  General  Synod  were  as  follows  : 
Van  Bunschooten  fund,  1 14730  ;  professoral  fund,  $120.52 
'57;  permanent  fund,  S9133-05:  how  many  Hebrew  mi nae 
of  sdver  is  that  ? 

9  2.  How  many 'drachmae  of  gold  is  the  annual  interest  at 
7  per  cent  ? 

923.  One  of  their  theological  professors  has  $1750  a  year, 
and  the  other  ^ . 5 sO  ;  how  many  lioinin  asses  in  the  whole  > 

924.  In  1821,  the  Board  of  Missions  of  the  General  As- 
sembly appointed  missionaries  to  labor  m  the  destit  iie  set- 
tlements, 168  weeks  ;  how  many  Attic  minse  of  silver  would 
th;it  ain  unt  to  at  %)  a  week  ? 

9  3.  The  collections  reported  for  missionary  purposes, 
amounted  to  S^'l^'i  51  ;  how  many  weeks  labor  would  that 
pay  for,  at  %9  a  week  ? 

9  26.  The  collections  reported  for  the  Commissioners'  fund, 
^iMC  3i4t6*24  ;  how  many  miles  travel  would  that  pay  for, 
at  o  J  cents  per  mile  ? 

9^7.  In  1821,  the  permanent  fund  of  the  American  Educa- 
ti.'ni  8.>ciety  was  S  t)87o*50  :  how  many  rials  of  velion  in 
tueaiii'Ml  interest  at  6  per  cent  •' 

9:is  In  i'3  K  the  receipts  of  the  Young  Men's  Missionary 
Society  of  N.  Vork,  were  S237j'77;  how  many  pagodas  is 
that  ?  ' 

,    9-29.  In    1813,   the    receipts  of  the  Hampshire  Miss.  SoCc 
were  %1527''25  ;  how  many  sequins  is  that  ? 

9  >0.  The  Leeds  and  Liverpool  canal  is  139  ndles  long, 
and  cost  800  .'wO/.  sterling  ;  what  iS  that  per  mile  ? 

931.  Tb^  canal  of  Languedoc  is  180  miles  long,  and  cost 
Xd 40000  ;  what  is  that  per  mile  ? 

932.  The  Middlesex  canal,  in  Massachusetts,  is  28  miles 
long,  and  cost  4780J0  dolls. ;  what  is  that  per  mile  r 

933.  rhe  Erie  canal,  in  New  York,  is  expected  to  be  36S 
mUes  long,  and  to  cost  457 1814  dolls. ;  what  will  that  be  per 
mile  ? 

934.  Tf  one  million  of  tons  should  pass  through  it  annually, 
and  the  tt)ll  should  be  a  cent  and  a  half  per  ton  per  nule;, 
what  will  be  the  amount  ? 


Qiiestions.  2\T 

5.  If  only  half  that  amount  be  produced,  and  S500000 
be  annually  required  for  repairs,  officers,  &c.  5  what  will  be 
the  annual  net  proceeds  to  the  s'ate  - 

936.  If  the  whole  expense  should  be  at  simple  interest  at 
7  per  cent,  three  years  before  the  canal  is  opened,  and  one 
year  more  before  a  year's  toll  can  be  applied  to  pay  it,  and. 
tho  net  proceeds  be  applied,  at  the  end  of  every  year,  to  ex- 
tinguish the  debt ;  how  much  will  re  n:iin  to  the  state  at  the 
end  of  the  third  year  afier  the  canal  is  opened  ? 

937.  If  500000  tons  pass  through  the  canal  annually,  and 
the  transportation  by  land  was  90  dolls,  per  ton,  and  on  the 
canal  is  S^'^J^,  and  the  toll  is  as  stated  a'jqve  ;  what  is  the 
annual  saving" to  the  public? 

938.  If  the  toll  and  transportation  together  are  2  and  a 
.half  cents  per  mile,  and  a  million  of  tons  sli  )uld  be  annuiUy 

transported  on  the  canal  ;  what  would  be  the  annual  saving 
^     to  the  public,  on  eveyj  mile  the  canal  shouhl  be  shortened  ? 

939.  It  is  estimated,  that  if  tlie  canal  cv)uld  be  made  di- 
rectly from  Schenectady  to  x\lbany,   instead  of  going  round 

,     by  the  Cohoes,  the  distance  would  be  shortened  I >  mdes,  and 

I  ^  other  ijdvanta.^es  gained  e'[ual  to  a  saving  of  5  miles  more  ; 

Z    if  so,  what  vvould  be  the  annual  saving  to  the  public,  on  toll 

'  and  transportation  alone;  and    wiiat  capital   would   this  be 

equivalent  to,  on  the  principle  of  perpetuities,  discounting  at 

5  per  cent  compound  interest  ? 

940.  If  the  width  of  the  canal  is  28  feet  at  the  bottom,  and 
40  feet  at  the  top  of  the  water,  and  the  water  is  4  feet  deep; 
how  many  cubic  feet  of  water  are  in  every  foot  in  length  ?  • 

941.  In  1316,  the  permanent  annual  expenses  of  the  go- 
vernment of  the  United  States,  were  estiuiated,  by  the  Se- 
cretary of  the  Treasury,   to  be  i^3o00u00  doils.  ;  of  which, 

:     1765313  dolls,  were  civil,    diplomatic,   and    miscellaneous; 
64596^6  dolls,   were  military;  S9866o9  dolls,   were  naval  ; 
28^3:20  i  dolls,  were  incidental  ;  and  the  rest  to  pay  the  inte- 
rest, and  reduce  the  prnicipal  of  the  public  debt :  how  much  „ 
'is  the  last  ? 

94^2.  It  is  estimated,  tliat  the  annual  expense  of  militia 
trainings  in  the  United  States,  i,  5  millions  of  dollars.  Add 
that  to  t!\e  uulitary  and  naval  expenses  of  the  government,  a» 
above  stated,  and  tell  the  asnount. 

943.   If  that  amount  shouhl  be  appropriated  to  spread  th« 
gospel  of  pea  e,  how  many  bibles  would  one  eighth  of  it  aii- 
iiualiy  furnish,  at  60  cents  each  ? 
T 


218  Questions. 

944.  How  many  missionaries  would  one  fourth  of  it  sup- 
port, at  500  dolls,  each  ? 

945.  How  many  young  men  would  one  eighth  of  it  assist 
in  their  education  for  the  ministry,  at  1 25  dolls,  each  ? 

946.  How  many  tracts,  of  12  pages  each,  would  one  six- 
teenth of  it  annually  furnish,  at  one  mill  per  page  ? 

947.  How  many  children  would  the  rest  of  it  keep  at 
school,  at  the  rate  mentioned  in  No.  b  i  7  ? 

948.  Congress  reserved  of  the  public  lands  in  Alabama, 
460u0  acres  for  a  university,  which  is  estimated  at  li  dolls, 
per  acre  ;  if  it  should  be  sold  for  that,  and  the  money  placed 
in  a  fund  which  yields  6  per  cent,  what  will  be  the  yearly 
amount  ? 

94i^.  The  representation  in  Congress,  according  to  the 
sensus  of  i8£0,  is  to  be  as  follows  :  New- York,  34  represen- 
tatives; Pennsylvania,  26;  Ohio,  14;  Massachusetts,  i3; 
Maine,  7  ;  Connecticut,  6  ;  NewJersey,  6  ;  New-H  impshire, 
6  ;  Vermont,  5  ;  Indiana,  3  ;  Rhode-Island,  2  ;  Delaware,  1 ; 
Illinois,  1  :  how  many  in  the  Northern  states  ' 

950.  In  the  Southern  states,  Virginia,  .2  ;  North- Carolina, 
13  ;  Kentucky,  12;  Tennessee,  9  ;  feouth-Carolina,  9  *,  Mary- 
land, 9  ;  Georgia,  7 ;  Louisiana,  3  ;  Alabama,  3  ;  Missouri, 
1  ;  Mississippi,  1  :  how  manj  in  these  ? 

961.  How  many  in  all  ? 

952.  'Ihe  value  of  domestic  articles  exported  from  the 
United  States  in  the  ;^ear  1819,  was  as  follows  :  Produce  of 
the  sea,  82024000  ;  of  the  forest,  ^4927000  ;  of  agriculture, 
g4 140  2000 ;  manufactures,  S-*;dV4lOu  ;  uncertain,  gbSOOOO ; 
how  much  in  all  / 

953.  The  value  of  foreign  articles  exported,  was  18OC8029 
dollars;  what  is  the  total  value  of  exports  in  that  year? 

964.  If  one  per  cent  of  this  was  devoted  to  charitable  pur- 
poses, how  many  livres  would  it  be  ? 

955.  The  mean  temperature  of  Boston  in  1819,  according 
to  observations  made  three  times  a  da}?,  was  as  follows  :  Ja- 
nuary, 30  degrees  of  Fahrenheit's  theruiometer ;  February, 
31  ;  March,  29;  April,  41  ;  May,  52 ;  June,  67  ;  July,  71  ; 
August,  69;  September,  64;  October,  53 ;  November,  40'; 
December,  31 :  what  was  the  mean  temperature  of  the  whole 
year  ? 

956.  The  number  of  inches  of  rain  which  fell,  was  as  fol- 
lows :  January,  1'05  ;  February,  2-27  ;  March,  6*51  ;  April, 
3-74;  May,  3-06;  June,   3-o6;  July,  2-02;  August,  4*38 ; 


Questions.  ^  219 

September,  5'27  ;  October,  1-40;  November,  1'22;  Decem- 
ber, 1*29  :  how  many  inches  in  the  year  ? 

957.  From  July  18:6  to  June  1817,  the  mean  degrees  of 
heat  at  Bombay,  were  as  follows  :  July,  80  ;  August,  78i  ; 
September,  79i  ;  October,  S:.^  ;  November,  82 J  ;  December, 
79J  ;  Januarv,  78^  ;  February,  76i  ;  March,  79  ;  April,  h3| ; 
May,  86^  ;  /une,  82^  :  what  was  the  mean  heat  of  the  whole 
year  ? 

958.  The  population  of  Massachusetts  in  1820,  was  stated 
as  follows  ? 

males.  females. 

Whites,  under  ten  years  old,  70993  69265 

from  ten  to  sixteen,  38573  38303 

from  16  to  twenty -six,  49506  52805 

from  26  to  forty  five,  544U  57721 

from  45  upwards,  ^^'6^^  46171 

Blacks,  3308  3560 

What  is  the  whole  number,  and  how  many  more  females  than 

males  ? 

959.  In  ^819,  the  number  of  revolutionary  pensioners  was 
stated  as  follows:  In  N.  Hampshire,  1142;  Maine,  1824; 
Massachusetts,  25l4  ;  R,  Island,  249;  Connecticut,  1373  j 
Vermont,  1296:  how  many  in  New  England? 

960.  In  N.  York,  3196  ;  N.  Jersey,  467  ;  Pennsylvania, 
1090;  Delaware,  41;  Maryland,  575;  Dist.  of  Columbia, 
51  :  how  many  in  the  Middle  states  ? 

961.  In  Virginia,  69i;  N.  Carolina,  212;  S.Carolina, 
180;  Georgia,  46  ;  Alabama,  5  ;  Mississippi,  6;  Louisiana, 
1  :  how  many  in  the  Southern  states  ? 

962.  In  Kentucky,  474  ;  Tennessee,  114  ;  Ohio,  647  ;  In- 
diana,  96  ;  Illinois,  4  ;  Missouri,  6  ;  Michigan,  3  :  how  many 
in  the  Wes  ern  states  ? 

963.  How  many  in  all  the  states  ? 

964.  What  would  be  the  amount  of  pensions,  at  8  dollars 
a  month  each  ?  * 

9:>5.  The  amount  of  the  funded  debt  of  Great  Britain  and 
Ireland  in  1.^12,  was  £906939589  ..  16 ..  8,  and  unfunded, 
=g5639r848  ..  16..  10;  what  is  the  whole  in  federal  money  ? 

966.  What  was  the  value  of  the  golden  candlestick  and 
its  appendages,  being  a  talent  of  gold  ? 

9S7.  What  was  the  whole  value  of  the  gold  and  silver  of 
the  tabernacle,  being  29  talents,  730  shekcb  of^old,ancl  100 
talents,  1775  shekels  of  silver  ? 


220  Quest  iQUs. 

968.  What  was  the  value  of  a  chariot  from  Egypt,  in  Sor- 
lomon's  time,  being  6(0  shekels  of  silver  ? 

969.  What,  of  a  horse,  being  150  shekels  of  silver  ? 

970.  If  the  price  of  redemption  of  a  field  sown  with  a 
homer  of  barley,  was  50  shekelj*  of  silver,  (Lev.  27.  l6,)  what 
was  that,  in  federal  money,  for  the  ground  sow  n  w  ith  a  bushel? 

97 1.  How  many  quarts,  dry  measure,  w^as  the  allowance 
©f  manna  for  each  person,  being  an  omer  ? 

972.  How  many  galkms  of  wine  did  our  Lord  make  by  his 
miracle  at  Cana,  if  each  of  the  6  water  pots  contained  2  J 
baths  ? 

973.  What  was  the  value,  in  federal  money,  of  the  oint- 
ment which  Mary  poured  on  the  feet  of  Jesus,  being  SOO 
Roman  pence,  or  denarii  i* 

974.  How  long  is  the  side  of  a  cube  of  fine  gold,  which 
weighs  a  ton  ? 

975.  What  is  the  solid  content  of  2  Cwt.  of  steel  ? 

976.  if  the  diameter  of  the  Earth  is  7928  miles,  and  that 
of  the  Sun,  885248  ;  how  many  times  larger  than  the  Earth 
is  the  Sun? 

977.  How  many  wine  gallons  in  10  C«vt.  of  proof  spirits  ? 

978.  How  many  solid  inches  in  a  living  man  who  weighs 
170  lb.  avoirdupois  ? 

979.  A  and  B  travel  the  same  way,  as  follows  :  A,  55 
miles,  6  furlongs,  27  poles  ;  S6  m, ;  27  m.  28  p.  ;  29  m.  3 
fur. :  B,  36  m.  3  fur.  29  p  ;  i^7  m. ;  28  m.  27  p. ;  34  m.  7 
fur.  :  and  then  A  travels  back  24  m.  3  fur-  i7  p. ;  24  m.  6 
fur. :  liow  far  asunder  are  they  ? 

98u.  There  are  two  numbers,  the  less  is  8967,  and  their 
difference  three  times  as  many  ;  what  is  the  greater  number? 

981.  There  are  two  numbers,  the  greater  of  which  is  79 
times  209,  and  their  difference  2t)  tmies  ^9  ;  what  is  their 
sum  ? 

982.  An  apothecary  mixed  5  sorts  of  medicines,  each  3  lb. 

11  oz.  7  dr.  2  sc.  13  gr. ;  and  7  sorts,  each  11  oz.  5  dr.   i  sc. 

12  gr. ;  how^  much  in  all  ? 

983.  What  is  the  amount  of  twice  twenty  five  added  to 
tv/ice  five  and  twenty  ?  ^ 

984.  Bought  of  A,  S?' gol.  3  qt.  of  wine  ;  of  B,  three  times 
as  much,  and  .^  g^irj^r^.  more  ;  of  C,  as  much  as  of  A  and 
B  both,  and  7  gal.  3  qt,  1  pt.  more  :  sold  T>,  7  gal-  1  pt.  ;  E, 
5  gal. ;  F,  as  much  as  the  difftrence  between  D  and  E,  and 
2  gal.  1  c^t,  more ;  how  much  is  unsold  ? 


985.  Add  333  eagles,  333  dolls.  333  dimes,  333  cents,  and 
333  mills,  together. 

■  986.  How  many  lb.  Troy,   in  f  of  |  of  i  of  4  of  ^^  off  oi 
800  lb.  avoirdupois? 
987.  What  day  of  the  year  was  the  last  day  of  June  1816'^ 

988.  By  what  must  Jl30..  8..  6^  be  divided,  to  give  .- 
quotient  of  ^6..  17..  31? 

989.  If  a  debtor  who  owes  L2000,  pays  but  ^625,  how 
much  is  that  on  the  pound  ? 

990.  Tell  the  product  of  i  off  off  off  of  1500,  by  -|  of 
,^j  of  90. 

991.  From  9  lb.  7  oz.  10  dwt.  of  silver,  how  many  spoons 
can  be  made,  each  weighing  2  oz.  15  dwt.  ? 

992.  How  many  solid  feet  in  a  ton  of  cork  ? 

993.  Bought  two  parcels  of  flour,  which  together  weighed 
19  Cwt.  3  qr.  4  lb.,  for  ^97..  17..  6;  their  difference  in 
weight  was  3  Cwt.  1  qr.  4  lb.,  and  in  price,  =g8  ..  13  ..  6  ; 
what  was  the  weight  and  value  of  the  greater  parcel  ? 

994.  Received  ^62000.',  and  paid  16  persons  ^54  .- 10.. "6 
feach,  and  3  persons  ^  1 93  ..  6  ..  4  each  ;  what  is  }  of  the  rest? 

995.  A  was  born  August  6,  17?^3,  and  B,  Oct.  21,  1815; 
how  old  will  A  be  when  B  comes  of  age  ? 

996.  There  are  two  numbers,  of  which  the  greater  is  76 
cimes  1 1 1,  and  their  difference  is  1 8  times  27  ;  what  is  th^ir 
product  ? 

997.  From  19  chests  of  tea,  each  3  Cwt.  2  qr.  7  lb.,  how 
many  canisters  of  7  lb.  each  can  be  filled  ? 

998.  Sent  to  the  bank  Si  eagles,  9|  dolls.  9^  dimes,  and 
V},  cents  ;  and  drew  clieoks  for  g.36-25,  g27-27^and  g  19-34 1 
how  much  is  left  ? 

999.  One  planet  has  moved  through  9  si2:ns,  29  degrees, 
£9  minutes,  25  seconds  of  its  orbit ;  and  another  has  moved 
5  signs,  »5  <lejrrees,  18  minutes,  3  seconds,  less  than  the  firsti 
how  much  does  the  latter  want  of  a  complete  revolution  ? 

1 000.  If  a  wheel  is  9  feet  2  inches  in  circumference,  how 
»iany  times  will  it  turn  round  in  running  150  miles? 

1001.  How  many  doT'.ens  of  gallon,  quart,  and  pint  bottles, 
of  each  an  equal  number,  maybe  filled  fiona  a  cask  of  wine 
containing  23:  gallons  ? 

1002.  Bought  125  Cwt.  2  qr.  gross  of  sugar,  tare  176  Ib,^ 
irett  4  lb.  per  iG4  lb  ;  how  many  lbs.  neat  ? 

100  .  How  much  did  1  gain,  if  1  sold  ^t  for  3|d.  ^^  3fe 
mmQ  than  1  gave  for  it  ^ 


^12^  (luesiioris. 

1004.  If  2 545  cost  ^236  ..  18  ..  6^  what  cost  1565  r 

1005.  What  sum,  at  interest  for  9  years  and  6  monthB, 
will  amount  to  L428  ..  5,  if  the  rate  per  cent  is  4§  ? 

1006.  How  long  is  the  side  of  a  cube  of  marble,  which 
weighs  a  ton  ? 

1007.  D  has  linen  worth  20d.  an  ell,  ready  money  ;  but  in 
barter  he  would  have  2s.  E  has  cloth  worth  14s.  6d.  per  yd* 
ready  money  ;  at  what  price  ought  it  to  be  rated  in  barter  ? 

1008.  A  man  left  his  son  ^1500,  to  receive  the  amount  at 
5  per  cent,  when  he  should  come  of  age,  which  was  then  found 
to  be  Z2193 ..  15;  how  old  was  the  boy  when  the  bequest 
was  made  ? 

1009.  If  a. man  can  travel  300  miles  in  18  days,  when  the 
days  are  14  hours  long;  in  how  many  days  can  he  travel 
950  miles,  when  the  days  are  but  12  hours  long  ? 

10  iO.  Divide  360  into  4  parts  which  shall  have  the  ratio  of 
3,  4,  5,  and  6. 

1011.  A  owes  B  a  sum  of  money,  of  which  §  is  payable  in 
2  months,  i  in  4  months,  |  in  6  months,  and  4  in  8  months  ; 
what  is  the  equated  time  ^ 

101*3.  Tell  the  amount  of  jL50,  at  compound  interest,  for 
"^  years,  at  5  per  cent,  the  interest  payable  half  yearly  ? 

1013.  Tell  the  amount,  if  payable  quarterly  ? 

1014.  A,  B,  and  C,  trade  together.  A  at  first  put  in  480 
dolls,  for  8  months,  and  then  put  in  200  dolls,  more,  and 
continued  the  whole  ^  months  longer,  and  then  took  out  his 
whole  stock.  B  put  in  800  dolls,  for  9  months,  and  then 
iook  out  I)583'333,  and  continued  the  rest  3  months  longer. 
C  put  in  D366-666  for  10  months,  then  put  in  D250  more,  and 
continued  the  whole  6  months  longer.  At  the  end  of  their 
partnership,  they  had  cleared  DlOOO  ;  what  is  each  man's 
share  ? 

1015.  How  many  ale  gallons  in  1  Cwt.  of  cow's  milk? 

1016.  What  cost  648  A.  2  R,  36  p.  at  £2.,  19..  11^  per 
acie? 

1017.  If  my  income  is  500  English  guineas  a  year,  and  I 
spend  IBs.  N.  England  currency,  a  day  ;  in  what  time  shall 
I  save  3715  dollars  ? 

1018.  How  many  yds.  of  matting,  that  is  §  yd.  wide,  will 
©over  a  floor  that  is  12  feet  by  28  *? 

1019.  Suppose  1 000  men  in  garrison,  with  provisions  suffi- 
cient for  3  months ;  how  many  must  depart,  that  the  provi- 
mm  may  last  15  months  ? 


^uestidu$*  228 

S^O.  A  certain  cistern  has  5  pipes ;  by  the  first  alone,  it 
can  be  filled  in  1 2  minutes,  by  the  second  in  1 5,  by  the  third 
in  20,  by  the  fourth  in  30,  and  by  the  fifth  in  60  ;  in  \vhat 
time  will  all  running  together  fill  it  ? 

1021.  What  is  the  weight  of  a  hhd.  of  rain  water  ? 

1022.  If  the  earth  moves  round  the  sun,  596900000  miles, 
in  335i  days,  how  far  are  we  carried  per  second  by  this  mo- 
tion ? 

1023.  How  far  are  those  who  live  under  the  equator,  car- 
ried per  hour,  by  the  diurnal  motion  of  the  earth  ? 

1024.  If  sound  flies  1 142  feet  per  second,  and  light  is  seen 
instantaneously;  and  if  a  person's  pulse  beats  70  timts  a 
minute,  how  far  distant  is  the  explosion  of  a  clap  of  thunder, 
when  you  count  8  pulsations  between  seeing  the  flash  afid 
hearing  the  report  ? 

1025.  How  much  in  length,  that  is  24  poles  wide,  will 
make  2  acres  ? 

10£6.  Received  from  Jamaica,  28  hhds.  sugar,  each  12 
Cwt.  1  qr.  10  lb.,  their  Cwt.  being  100  lb. ;  how  many  of  our 
Cwt.  are  in  the  whole  ? 

1027.  Boiight  a  parcel  of  cloth,  at  the  rate  of  12s.  8d.  for 
every  2  yds. ;  and  sold  a. quantity  at  the  rate  of  45s.  lOd.  for 
every  3  yds.,  by  which  as  much  was  gained  as  150  yds.  cost; 
how  much  was  sold  ? 

1028.  Lent  250  dolls.,  and  at  the  end  of  8  months  received 
'260  ;  at  what  rate  was  the  interest  computed  ? 

1029.  If  a  staff  4  ft.  6  in.  long,  standing  erect,  casts  a  sha- 
dow, at  12  o'clock,  7  ft.  4  in. ;  how  wide  is  a  ditch,  running 
due  east  and  west,  at  the  north  side  of  a  wall,  which  is  71  ft. 
high,  and  stands  2 1  ft.  from  the  ditch,  and  the  shadow  of  the 
v/all,  at  12  o'clock,  reaches  13  ft.  8  in.  beyond  the  ditch  ? 

1030.  If  4  ells  Flemish  cost  L\  ..  6  ..  8,  what  must  be  paid 
for  10  pieces,  each  containing  21  ells  English  ? 

1031.  If  68  gallons  of  water  fall  into  a  cistern  per  hour, 
and  24  run  out,  and  the  cistern  contains  4  hUds. ;  in  what 
time  will  it  be  filled  ? 

1032.  In  the  year  1768,  the  parish  of  S.  F.  in  the  state  of 
Connecticut,  gave  a  call  to  a  minister  of  the  gospel,  and  of- 
fered him  175/.  settlement,  and  70/.  a  year  salary,  N.  Eng. 
currency;  and  stated  the  price  of  wheat  at  4s.  a  bushel, 
rye  2s.  8d.,  and  corn  !2s.  How  many  dolls,  salary  would  be 
equal  to  that,  when  wheat  is  12s.  a  bushel,  rye  6s.,  and  corn 
^s.  6d.,  the  salary  being  increased  by  the  amount  ©f  the  set- 


224  Questions* 

tlement,  on  the  principle  of  annuities,  discounting  at  5  per 
cent,  compound  interest,  and  reckoning  25  years  the  life  of  a 
minister  ? 

1033,  A  man  has  two  sons,  A  and  B.  At  the  age  of  14, 
he  tells  them  that  he  will  give  them  2uO0  dolls,  each,  and 
allow  them  to  choose  their  profession.  A  chooses  to  have  a 
liberal  education,  and  be  a  minister  of  the  gospel.  B  chooses 
to  be  a  blacksmith.  A's  money  is  put  out  at  interest,  at  6 
per  cent;  and  he  draws  upon  it,  from  year  to  year,  to  defray 
the  expense  of  his  education  ;  in  obtaining  which,  he  spends 
8  years,  and  exhausts  his  fund.  B's  money  is  put  out  at  in- 
terest, at  the  same  rate  ;  but  having  no  occasion  to  draw 
upon  it,  the  interest  is  every  year  added  lo  the  pincipal. 
At  the  age  of  25,  they  both  settle  in  the  same  parish  ;  A  as 
their  minister,  B  as  their  blacksmith.  The  people  support 
A,  so  that  he  provides  comfortably  for  himself  and  family, 
and  lays  up  50  dolls,  ev^ry  year.  By  industry  in  his  busi- 
ness, B  supports  himself  and  family  equally  well,  and  lays 
up  the  same  sum.  At  the  end  of  42  years  from  their  settle- 
ment, they  die,  leaving  to  their  families  the  same  amount  of 
property  saved  from  their  earnings.  But  in  addition  to  this, 
B  has  his  2000  dolls,  with  its^compound  interest  since  he  was 
14  years  old,  which  A  might  have  had  also  if  he  had  chosen 
the  same  profession.  How  much  has  A  relinquished,  and 
really  given  to  ius  parish,  for  the  privilege  of  being  their 
minister,  which  he  might  have  saved  to  himself,  if  he  had 
been  their  blacksmith  ? 

1C34«  Tell  th^  interest  of  2731^  for  one  year,  at  3^  per 
cent. 

1035.  There  are  two  numbers,   the  sum   of  which  is  90, 
and  their  pioduct  is    000  ;  what  are  the  numbers  ? 
.   1036,  How  many  men  should  reap  417'6  acres  in  12  days, 
when  5  men  reap  }  of  that  quantity  in  |  the  time  ? 

1037.  If  a  cellar  '■12-5  feet  long,  17-3  wide,  and  lo-£5 
deep,  is  dug  in  ^J  days,  by  12  men,  working  12*3  hours  a 
day ;  how  many  days  of  8*2  hours,  ^should  18  men  take  to 
dig  one  45  feet  long,  34'6  wide,  and  12*3  deep  ? 

1 038.  How  vide  is  a  street,  when  two  ladders,  each  30 
feet  long,  placed  foot  to  foot,  reach,  one  to  a  window  16  feet 
high  on  one  side  of  the  street,  and  the  other  to  a  window  20 
feet  hi^h  on  the  other  side  ? 

1©39.  In  600Z.  Canada,  how  much  S.  Carolina  ? 
1040.  How  much  in  length,  of  apiece  of  land  that  k  11J|^ 
poles  v/ide,  will  make  an  acre  ? 


Questions.  -225 

1041.  Reduce  f  of  f  of  J  of  4,  to  a  single  fraction. 
10i2.  Multiply  -♦  ft.  7  in,  by  6  ft.  4  in. 
104.3.  Reduce  f  of  a  pole  to  the  fraction  of  an  acre. 
.  lo44.  Find  the  difterence  betwiien  f  of  i Of,  and  |  of  20. 
-     1045.  In  600/.  Canada,  how  much  Irish  ? 

1046.  Find  the  diiference  between  |  of  f  of  19,  and  -J  of 
I  of  23/^. 

1047.  Reduce  |,  2f ,  and  4,  to  a  common  denominator. 

1048.  Reduce  f  of  a  penny  sterling  to  the  fraction  of  an 
.  Enii;lish  guinea  ? 

^  1049.  How  many  square  yds.  of  carpet  will  cover  a  floor 
28  ft.  by  16  ? 

1050.  Reduce  j\  of  a  barley  corn  to  the  fraction  of  a  mile. 

105  1.  How  much  in  length,  that  is  7 1  inches  wide,  will 
make  a  square  foot  ? 

1052.  A  line  28  yds.  long,  will  reach  from  the  top  of  a  wall 
28  ft.  high,  standing  on  the  brink  of  a  ditch,  to  the  opposite 
bank ;  how  wide  is  the  ditch  ? 

10  3.  Reduce  |  of  a  nail,  to  the  fraction  of  an  ell  English. 

1054.  Reduce  |  of  alb.  Troy,  to  its  value. 

10  5.  Multiply  -385740  by  -00464. 

1056.  Reduce  ^V  of  a  grain,  to  the  fraction  of  a  lb.  Troy. 

1()>7.  in  40i)l.  sterling,  how  much  Maryland? 

1058.  Reduce  44V25  to  its  lowest  terms. 

1059.  Tell  the  t^quare  root  of  jVt- 

1060.  Reduce  |  of  a  penny  ster.  to  the  fraction  of  a  dollar. 

1061.  Find  the  difterence  between  ^  of  13^2_^  ^n^j  j  of  ^  of 
of  67^5_, 

1062.  Reduce  |d.  N.  York,  to  the  fraction  of  an  English 
guinea. 

1063.  Tell  the  difterence  between  f  of  a  lb.  avoirdupois, 
and  If  of  an  ounce. 

1064.  Tell  the  square  root  of  jV 

1065.  What  is  the  4th  root  of  4  96. 

1066.  Tell  the  difterence  between  H^  aind  y^^s, 

1067.  Tell  the  difference  in  sterling  between  f/.  sterling, 
and  f  of  a  dollar. 

1068.  What  is  the  value  of  r^^  of  a  year  ? 

1069.  When  6  persons  use  1|  lb.  of  tea  in  2  months,  how 
much  will  suftice  8  persons  |  a  year  ? 

107  .  Reduce  y3j  of  a  day  to  its  value. 

1(^71.  Find  the  cube  root  of  the  square  root  of  262144, 

1072,  Tell  the  difference  between  2714  and -916, 


^26  ^uestions^ 

1073.  In  100/.  Irish,  how  much  Connecticut? 

1074.  Tell  the  sum  of  i^T6+54-321  +  li2+'65+l2-5  + 
•0463. 

1075.  Reduce  f  of  a  crown  to  the  fraction  of  an  English 
guinea. 

1076.  Divide  234-70525  by  64-25. 

1077.  In  lOOZ.  Iri^h,  how  much  Georgia? 

1078.  Reduce  f  of  i  crown  to  the  fraction  of  an  English 
shilling. 

1079.  Tell  the  difference  between  |  of  2s.  6d.  &  f  of  5s.  8d. 

1080.  Divide  14  by  -7854, 

1081.  In  IGOL  Irish,  how  much  sterling? 

1082.  How  many  feet  is  the  side  of  a  square  containing 
^11  of  a  square  mile  ? 

1083.  Reduce  3  ft.  8  in.  to  the  fraction  of  a  mile. 

1084.  Reduce  J  of  ^  of  |,  &  3^,  to  a  common  denominator. 
108.5.  Divide  5-1 6  by  1000. 

1086.  Reduce  3  qt.  li  pt.  to  the  fraction  of  a  hhd. 

1087.  Tell  the  difference,  in  federal  money,  between  f  of 
a  guilder,  and  ^  of  3^  livres. 

1088.  Tell  the  difference,  in  federal  money,  between  2/. 
lis.  N.  England,  and  |  off  of  lof  dolls. 

IoSj.  If  2  ships  sail  from  the  same  port,  one  north  76 
leagues,  the  other  east  58  leagues  ;  how  far  are  they  asunder? 

1090.  Reduce  4  lb.  3  oz.  6  dr.  to  the  fraction  of  a  Cwt. 

1091.  Divide  6  by -6. 

1092.  Tell  the  value  of -3375  of  an  acre. 

1093.  Reduce  3s.  5Jd.  ster.  to  the  fraction  of  a  guinea. 

1094.  In  4oo  livres,  how  mucn  Nova-Scotia? 

1095.  Add  ^L  fs.  and  yVl. 

1096.  Add  J,  f ,  1,  and  |  of  f  of  5. 

1097.  Reduce  ^//^  to  a  decimal. 

1098.  In  4oo  livres,  how  much  Irish  ? 

lo9i:'.  Tell  the  difference  between  5^,  and  f  of  4^. 

11 00.  Tell  the  deference  between  |  off  of  7  Cwt.  3  qr. 
12  lb.,  and  ^  ot  J  of  j%  of  5  Cwt.  3  qr.  27  lb. 

1  lol.  Tell  the  difference  between  ^l.  and  |  of  fs. 

1  lo2.   Reduce  -6875  yd.  to  its  value. 

1103.  it  a  ship  of  -^oo  tons  has  8o  feet  keel,  what  must  be 
the  keel  of  another  of  the  same  shape,  to  carry    5o  tons  ? 

1  lo4.  Add  7|  off,  and  f  of  4  of  7.  and  5f,  and  j\. 

1  loi.  FinrI  the  side  of  a  cubical  bi»x  containing  12  bushels. 

II06.  Add  ^s.  and  1^. 


Questions.  227 


m 

HI  ilo7.  Add  f  of  a  farthing,  and  j\  of  a  shilling. 

lloS.    i'cll  the  difference  between  f  of  o^L,  and  |s. 

I  lo9.  In  600  guilders,  how  much  Canada  ? 
11,0.  Add  f  dwt-  and  j\ lb.  Troy. 

111'.  What  cost  564,  at  6/.  13s.  4d.  each  ? 

1112.  Add  ^L  ^s.  and  fq. 

1113.  Tell  the  difference,  in  sterling:,  between  j%  of  |f  of 
5  Eng.  guineas  and  |  of  4  ot  }  off  of  2o  crowns. 

li  .  k  If  a  gh)be  of  sUver,  5  inches  in  diameter,  is  worth 
465  iolls.,  what  is  the  value  of  one  2  feet  in  diameter  ? 

I I  lo.  Reduce  '056  of  a  pole  to  the  decimal  of  an  acre. 
1116.  In  600  guilders, how  much  Irish  ? 

1 1  i7.  Multiply  |,  3^,  5,  and  f  off,  continually  together. 

1    . 8.  Ketiuce  -21  pt.  to  the  decimal  of  a  peck, 

n  i9.  Add  I  of  a  year,  f  of  a  day,  and  |  of  an  hour. 

1 1^0.  Reduce  1 1  minuter  to  the  decimal  of  a  day. 

1 1- I  Multiply  J,  f ,  and  4y\,  continually  together. 

1  ££.  Add  I  of  a  crown  to  |  of  a  dollar. 

1  2.>.  Add  f  hlid.  and  f  gal. 

1124.  Multiply  |-  by  |  off. 

1  2o.  Divide  |  of  ^  by  4  of  7|. 
11.6.  Add  ^L  ster.  to  f  of  an  Eng.  guinea. 

112,.  Add  |/.  sterling,  |  of  an  Eng.  guinea,  f  of  a  crowB^ 
and  I  of  a  dollar. 

1 128.  What  cost  12Cwt.  ?qr.  14lb.  at  7L  lo  s.  9d.  per  Cwt.? 

1129.  Reduce  I  of}  of  i  of  lo,  to  a  single  fraction. 
I  >3o.  In  5|s  N.  York,  5|s.  Vermont,  5§^.  M.Jersey,  5|s. 

eorgia,  5|s.  sterling,  and  6|  dolls.,  how  many  dollars  ? 

1131.  Tell  the  difference  between  f  of  /y  of  lo  inches, 
and  I  of  I  of  ,^0  ot  0  feet. 

il;i2.  What  cost  9  Cwt.  2  qr.  26  lb.  at  71.  los.  9d.  per 
Cwt.  ? 

1  1:j3.  Reduce  J  of  f  of  f  of  |  of  7,  to  a  single  fraction. 

1134.  In  |4.  ster.,  :a*31s.  Vermont,  and  g5i,  how  many 
dollars  ? 

1 135.  Reduce  ^  of  |  of  i  of  7h  to  a  single  fraction. 

1 136.  Add  1-2347.  and  |s.  and  f  of  a  farthing. 

1137.  If  a  board  is  '7.3  of  a  foot  wide,  what  length  of  it 
will  make  24  square  feet  ? 

1138.  Tell  the  difference  between  f  off  of  21  hhds.,  and 
S-789  gallons. 

1139.  A  cubical  stone  contains  42873  solid  inches  ;  what 
is  the  superficial  content  of  one  of  its  sides  ? 


^28  Questio}i9' 

11 40,  At  41.  \Ts.  per  Cwt.  what  cost  17  lb«? 

1141.  Nedure  |  of  a  Cwt.  to  the  fraction  of  an  ounce* 
11-2.  Tell  the  ditfeience  between^  of  4'689  yds.,  and  y\ 

of  f  of  iir^*67  nails. 

114:3.  Multiply  5,  |.,  #  of  |,  and  4J,  continually  together. 

1  i  44.  Multiply  I  of  I  of  1 1  j\,  by  ?  of  |  of  f  of  2o. 

1145.  Multiply  23-678  by  f  off  of  ^  of  l5f 

1  Kb.  Multiply  J  of  •/.  las.  6d.  by  f  of  J  of  19|. 

11 4 i'.  What  cost  o  hhd.  tobacco,  each  4  Cwt.  ^  qr.  7*4  lb., 
at  8s.  ^•3d.f.r4'£lb.  ? 

1148.  A  person,  after  spending  '»  and  i^,  and  ^\  of  his 
money,  has  l6o  dolls,  left  ;  how  nvuch  had  he  at  Hist  ? 

1  149.   Multiply  I  of  f  off  ot  ;  5  dolls,  by    .-7689. 

I  l5o.  Suppose  3oo  stones  were  laid  3  v(l^.  from  each  other 
in  a  line,  and  a  basket  was  placed  yds.  from  \he  first ;  how 
fiir  nuist  a  person  travel,  togath^-r  them  one  by  one  into  trie 
baket? 

llol.  Multiply  I  of  5  crowns  by  3-6789,  and  tell  the 
amount  in  sterling. 

1132.  A  person,  after  spending  i  of  |  off  of  his  monej, 
and  \  of  J  off  of  the  remainder,  has  Sdo  dolls,  left;  what 
hud  he  at  first  r 

I I  y3,  Mul  ipJy  I  of  1  of  -?-  of  s  Cwt-  3  qr.  £6  lb.  by  10-6o 

1 13  4.  Sold,  it*  j(U.  if  clo.h,  <he  tir^t  for  5s.,  and  the  last 
for  3/.  5-.,  in  anthnjelicai  progression  ;  what  wa^  the  com- 
mon  (hifereace  } 

I  I  55.  Multiply  j%  of  3/.  lo  s.  6d.  by  f  of  |  of  26-78. 

11 5b.  What  number  is  that,  of  which  |  of  |  of  |,  and  ^  of 
i  ^^  b»  ^»oount  to  2oo./ 

II  j7.  Divn'e  2oo  dolls  so  tliat  A  shall  have  twice  as  much 
as  13,  and  6  dolls,  more  ;  and  C  6  dolls,  moie  tlum  A  :  what 
is  each  man's  share  •'' 

li58  Tell  the  amount  of  an  annuity  of  S5o  dolls,  for  4 
years,  at  5|  per  cent,  compound  interest 

1 159.  Multiply  I  of  j\  of  3j  guineas  by  |  of  J  of  21,  and 
tell  the  a«nountin  sterling. 

1  16o.  A  person  being  asked  his  age,  said,  if  i  of  |  of  |  of 
the  years  I  have  lived,  be  multiplied  by  1^2,  and  the  product 
divided  by  6,  the  quotient  will  be  2o ;  how  obi  was  he  •' 

1  16 1.  Find  the  con  rent  of  a  triangle,  whose  sides  are  all 
equal,  and  together  lueasure  9o  chains. 

1162.  8^1(125  yds.  at  4d.  for  the  first,  8d.  for  the  secuud. 
Is.  for  the  third,  and  so  on  ;  what  was  the  last  ? 


(luestions.  229 

1 1  ^3.  Bous^ht  25  yds.  at  4d.  for  the  first,  8d.  for  the  second, 
I6d.  for  the  third,  and  so  on  ;  what  was  Ihe  last  ? 

1164.  Find  the  side  of  a  cubical  box  containing  1 7  bushels. 

llo5.  A  son  askhighi-;  father  how  old  he  was,  received  for 
answer,  your  age  is  now  J  of  mine,  but  9  years  ago  it  was  ^ 
of  mine ;  how  old  was  the  father  ? 

1 166.  Divide  f  of  J  of -j^^  of  5  Cwt.  by  |  of  i  of  28^V 

11*7.  A  stationer  sold  quills  at  15s.  a  thousand,  by  vhich 
he  cleared  §  of  that  money ;  afterwards ^e  raised  them  to  1 8s. 
a  thousand  :  what  did  he  gain  per  cent  by  the  latter  price  ? 

1 168.  Divide  i  of  -^\  of  -^^  of  50  acres,  by  |  of  f  of  37-25. 

1169.  Of  150^  expenses,  A  paid  10^  more  than  B,  and  C 
paid  half  as  much  as  A  and  B  both,  and  15/.  more ;  what  sum 
was  paid  by  each  ? 

1 170.  Tell  the  present  worth  of  an  annuity  of  50Lto  con- 
tinue 4  years,  discount  at  4  per  cent  compound  interest. 

1  71.  If  the  base  of  a  cylindrical  vessel  is  5  feet  in  diame- 
ter, how  high  must  it  be,  to  contain  50  bushels  ? 

1 172.  If  a  square  contains  250  acres,  how  long  is  one  side 
of  it? 

1173.  If  the  earth  were  a  perfect  sphere,  and  its  circum- 
ference 250  )0  miles,  how  many  square  miles  would  its  sur^ 
face  be  ? 

1174-  How  many  cubic  miles  would  it  contain  ? 

1175.  Divide  -3?  5  of  |f.  of  37-25    lbs.  Troy,  by  ^  of  1-23. 

1176.  How  many  difterent  numbers  often  figures  in  each, 
can  be  expressed  Vjy  our  ten  numeral  characters,  without  ha- 
zing the  same  character  twice  in  the  same  number  ? 

1177.  If  2-37^.  ster.  pay  for  21  bushels,  how  many  can  be 
purchased  for  |  of  |  of  24.67  guilders  ? 

1178.  if  a  cask,  the  head  diameter  of  which  is  27  inches, 
'ontains  1 13  gallons,  what  must  be  the  head  diameter  of  ano- 
her  of  the  same  shape,  to  contain  63  gallons  ? 

Ii79.  The  head  of  a  fish  is  12  inches  long,  and  its  tail  is 
alf  the  length  of  the  head  and  body  both,  and  the  body  is  6 
iches  longer  than  the  head  and  tail  both  ;  what  is  its  whole 
ngth  ? 

1180.  How  many  different  dozens  can  be  chosen  out  of  24 
idividuals  ? 
^     1181.  Find  the  area  of  a  triangle,  the  sides  of  which  mea- 
'''  ture  25  chains,  36  ch.  and  4  ?  ch. 

83.  If  ^  of  f  of  yV  «f  36|.  English  guineas  pay  for  J  of 
rf  of   34-567  yds.,  how  much  will  ^^^  of  ^  off  of  4567%^ 
l^^es  pay  for  ? 

U 


230  ^estions, 

1 1 83.  How  many  different  companies,  of  5  persons  each^ 
can  be  chosen  out  of  65  individuals  ? 

1 184.  Find  the  height  of  a  cylindrical  vessel,  whose  dia-^ 
meter  is    0  feet,  to  contain  100  bushels. 

1185.  Tell  the  amount  of  176^  for  4  years,  at  5  per  cent, 
compound  interest. 

i  '  86.  What  must  I  give  for  a  perpetuity  of  500  dollars, 
discounting  at  5|  percent  compound  interest? 

1  1h7.  If  -j^  of  I  of  J  of  5  yds.  cost  &67||  rubles,  how  many 
dollars  will  pay  for  23|^  yds.  ? 

1188.  Tell  the  compound  interest  of  235Z.  for  5  years,  at 
4  per  cent. 

1 189.  A  person  owned  4  of  a  ship,  and  sold  ^  of  his  share 
for  976/. ;  what  was  the  value  of  the  ship  ? 

1190.  Bought  goods  for  375  dollars,  and  sold  them  in  4 
months  for  476  dolls. ;  how  much  per  cent  per  annum  wa» 
gained  ? 

119U  If  f  off  of  375-63/.  pay  for  |of|of-j%  of3456f 
acres,  what  cost  |^  of  |  of  |  of  S76-78  acres  ?' 

1192.  Tell  the  area  of  a  circle,  whose  diameter  is  15  rods, 

1193.  Bought  goods  for  250  dolls,  ready  money,  and  sold 
them  for  345  dolls,  payable  in  10  months  ;  what  was  the  gam 
per  cent  in  ready  money,  supposing  the  discount  to  be  6  per 
cent? 

1194.  Bought  35  yds.  of  cloth  for  12/.  5s.,  of  which  part 
was  velvet,  at  9s.  per  yd.,  and  the  rest  linen,  at  4s.  per  yd. ; 
how  many  yards  of  each  ? 

1195.  What  perpetuity  can  be  purchased  for  200  dollars, 
discounting  at  5  per  cent  compound  interest  ? 

1196.  Tell  the  breadth  of  a  river,  according  to  Problem  7, 
of  Mensuration,  when  EF  is  26  rods,  FD  40  rods,  and  EC 
280  rods  ? 

1197.  Tell  the  area  of  a  circle  whose  diameter  is  25  chains, 

1198.  Find  the  solid  content  of  a  pyramid,  the  base  ofi 
which  is  a  triangle  having  each  side  30  feet,  and  the  perpen- 
dicular height  being  50  feet. 

1 199.  If  2  doz.  apples,  of  equal  size,  are  put  into  a  peck 
measure,  and  3  wine  pints  of  sand  just  till  it,  how  many  solid 
inches  does  each  apple  contain  ? 

1200.  A  person  being  asked  the  hour  of  the  day,  said,  the 
time  past  noon  is  equal  to  |  of  the  time  till  midnight ;  what, 
time  was  it  ? 

1201.  What  principal,  at  5  per  cent  compound  intere#. 
will  amount  to  520/.  18s.  7^d.  in  3  years  ? 


Questions.  231 

1  ^202.  If  the  wall  of  a  fortress  is  1 6  feet  high,  and  is  sur- 
rounded by  a  ditch  i^O  feet  broad,  how  long  must  a  ladder  be, 
to  leach  from  the  outside  of  the  ditch  to  the  top  of  the  wall  ? 

120>    Reduce  l5s.  6d.  to  the  decimal  of  a  pound. 

i204.  Tell  the  difference  between  -J  of  ^  of  J  ot  l6  yds. 
and  J  of  -f  f  of  4i  rods. 

1205.  If  a  ladder,  80  feet  long,  be  so  placed  as  to  reach  a 
window  40  feet  from  the  ground,  on  one  side  of  the  street, 
and  without  being  moved  at  the  foot,  will  reach  a  window 
22  feet  high  on  the  other  side,  how  wide  is  the  street? 

1206.  If  00  apples  were  placed  in  a  straight  line,  6  feet 
apart,  and  a  basket  placed  6  feet  from  the  first,  how  far  must 
a  person  travel,  to  gather  them  one  by  one  into  the  basket  ? 

120T.  A  merchant  sold  ?50  yds.  of  cloth,  at  Is.  for  the 
first  yd.  ^s.  for  the  second,  5s.  for  the  third,  and  so  on  ;  wha^t 
did  he  gain  or  lose,  if  he  gave  3/.  per  yd, 

1208.  If  9-5  yds.  cost  8^5-75,  what  cost  435-5  yds  ? 

1209.  Bought  >4  yds,  the  first  at  2  dollars,  and  the  last  at 
16 1  dollars,  in  arithmetical  progression  ;  what  was  the  com- 
mon difference  -^ 

12  0.  Tell  the  present  worth  of  an  annuity  of  S.50  dolls,  to 
continue  6  years,  at  7  per  cent,  simple  interest 

i  2  i  1.  Find  the  area  of  a  triangular  field,  of  vvhich  one  side 
measures  43  poles,  and  the  perpendicular  upon  it  15  poles 

1 2 1 2-  What  is  the  content  of  a  triangle,  whose  sides  are 
all  equal,  and  measure  34  rods  each  ? 

1213.  If  a  stick  of  round  timber,  whose  diameter  at  the 
butt  is  2  s  inches,  contains  100  solid  feet,  what  must  be  the 
butt  diameter  of  a  simil  r  stick,  to  contain  68  feet  ? 

1214  How  many  days  can  6  persons  seat  themselves  dif- 
ferently at  dinner  ? 

12  i  5-  If  a  triangle  whose  base  is  27  chains,  contains  35 
acres,  what  will  another  of  the  same  shape  contain,  whose 
base  is  50  chains  i* 

1216.  Mixed  10  bush,  wheat,  at  gl'25  a  bushel,  12  bush. 
rye  at  70  cents,  13  bush  corn,  at  60  cents,  20  bush,  barley, 
at  40  cents,  and  30  bush,  oats,  at  30  cents  :  what  is  a  bushel 
of  the  mixture  worth  ? 

1  17.  Find  the  content  of  a  four  sided  field,  which  being 
divided  into  two  triangles  by  a  diagonal  line,  that  line  meas- 
ures 41  chains,  and  the  perpendiculars  upon  it  from  the  op- 
posite angles  25  chains  and  19  chains. 

12i  8.  Find  the  content  of  a  field,  which  being  divided  in- 


^32  Questions. 

to  6  triangles,  the  sides  and  perpendiculars  upon  them  mea« 
ure  as  follows : 

Triangles.    Bases.    Perp.  Triangles.    Bases.    Perp. 

No.  1,      34  rods,  19  rods.  No.  4,      4.3  rods,  25  rods. 

2,      25  15  5,      49  S3 

S,       35  22  6,       50  34 

1219.  Mixed  the  following  quantities  of  sugar,  worth  the 
following  prices  per  cwt.  to  wit:  2  cwt.  at  9  dolls.  4  cwt.  at 
iO  dolls.  5  cwt  at  12  dolls,  and  6  cwt  at  14  dolls,  what  is 
S  cwt.  of  the  mixture  worth? 

1220.  Find  the  length  of  a  slanting  tree,  when  if  you  set 
up  a  pole  parallel  to  the  tree,  18  feet  long  from  the  ground, 
and  take  such  a  station  that  your  eye  is  in  range  with  the  top 
of  the  pole  and  the  top  of  the  tree,  and  also  in  a  range  with 
a  mark  on  the  pole  and  another  on  the  ix^ct,  each  5  feet  Irom 
the  ground,  your  station  is  i2  feet  from  the  pole,  and  55  from 
the  tree. 

12:;- 1.  If  £0  oz.  of  gold,  at  5 2.  per  oz.  12  lb.  of  silver,  at 
5L  10s.  per  lb.  and  50  lb.  of  copper,  at  5s  per  lb.  be  mixed, 
together ;  what  is  20  lb.  of  the  mixture  worth  ? 

12^2.  Find  the  present  worth  of  a  perpetuity  of  800  dolls, 
per  annum,  discounting  at  4  per  cent,  compound  interest. 

i  223.  How  long  must  be  the  side  of  a  cubical  box,  to  con- 
tain 20  bushels  ? 

1^24.  Tell  the  compound  interest  of  347  dolls  for  4  years, 
at  5  per  cent. 

1225.  Tell  the  number  of  solid  feet  in  a  stick  of  squired 
timber  which  is  l6  by  18  inches  in  diameter  throughout,  and 
65  feet  long. 

1226.  What  principal,  at  5  percent  compound  interest, 
will  amount  to  643t.  os-  !•  i778i.  in  6  years  ? 

1227.  Find  the  area  of  a  circle  whose  circumference  is 
340  rods. 

1228.  What  must  be  given  for  a  perpetuity  of  250  dolls. 
to  commence  in  6  years,  discount  at  5^  per  cent,  compound 
interest  ? 

1^29.  In  what  time  will  600  dolls,  amount  to  g71 4-6096, 
at  6  per  cent,  compound  interest  ? 

1230.  With  1 5  gals  brandy,  at  14s.  a  gal.  I  mixed  14 
gals,  of  whiskey  at  5s.  and  10  gals  of  water  at  0;  at  what 
rate  per  gal-  must  I  sell  it  to  gain  12  per  cent  ? 

1231.  A  man  has  a  square  garden,  one  side  of  which  mea- 
sures 23  rods,  and  a  circular  fish  ]>ond  in  the  center,  the  di- 
ameter of  which  ife  9  rods  ;  how  much  ground  has  he  ? 

1232.  How  much  gold,  at  4/.  per  oz.  and  silver,  at  12s. 


questions.  ^33 

per  oz.  must  be  mixed  with  20  lb.  of  copper  at  6s.  per  lb* 
that  60  lb  of  the  mixture  may  be  worth  ^240  ? 

1233.  Find  the  solid  content  of  a  pyramid  whose  base  is 
an  equilateral  triangle,  the  circumference  of  which  is  33  feet, 
and  the  perpendicular  height  40  feet. 

12  >4.  Which  is  the  most  valuable,  and  how  rpuch  so,  an 
annuity  of  ^300  for  6  years,  or  a  perpetuity  of  gSOO  to 
commence  after  6  years,  discount  at  5  per  cent  compound  in- 
terest ? 

I  23d.  At  what  rate  per  cent,  compound  interest,  will 
23  Id^  amount  to  -49^  l6s.  i    904d.  in  2  years  ? 

1236.  If  a  pyramid,  whose  perpendicular  height  is  15  feet, 
contains  192  solid  feet,  what  must  be  the  height  of  another 
of  the  same  shape,  to  contain  half  as  much  ? 

1237.  If  the  diameter  of  one  circle  is  :7  rods,  what  must 
be  the  diameter  of  another  to  contain  ^  as  much  ground? 

1238.  Twenty -two  persons  bestowed  charity  on  a  beggar; 
tlie  first  gave  id.  the  second  >d.  the  third,  7d,  and  so  on  ; 
what  did  the  last  one  give  ^ 

1239.  Two  faniil  es  set  out,  at  9  o'clock  in  the  forenoon, 
each  in  their  own  carriage,  to  go  to  a  place  20  miles  distant 
The  first  travels  at  the  rate  of  6  miles  an  hour,  and  the  other 
at  the  rate  of  5  miles  an  hour.  At  the  half  way  house,  the 
first  stops  15  minutes,  and  then  goes  on  to  the  end.  The 
second  stops  1  >  minutes  at  the  end  of  8  miles,  and  then  goes 
on.  When  the  first  gets  through,  it  stops  20  minutes,  and 
the  other  not  coming,  the  empty  carriage  goes  back,  at  the 
rate  of  8  miles  an  hour.  The  second  carriage  breaks  down 
at  the  end  of  12  miles,  and  after  a  delay  of  15  minutes,  the 
party  walk  forward  at  the  rate  of  4  miles  an  hour.  The 
first  carriage  meets  them,  and  takes  them  up,  an;l  proceeds 
with  them  at  the  rate  of  fi  miles  an  hour,  to  the  end.  At 
what  O'clock  does  the  second  family  reach  the  place  of  des- 
tination • 

1240  If  a  cone  15  feet  high,  contains  100  solid  feet,  how 
far  from  the  base  must  it  be  cut,  to  divide  it  into  two  equal 
parts  ? 

1^41.  Tell  the  difference  between  f  of  f  of  9f  acres, 
and  4  of  f  of  1  ^S^^^  poles. 

1242.  Multiply  -f  of  i  of  |  of  ''^f  of  25£.  \6s.  by  3-98. 

1243.  Divide  f  of  f  of  21,  by  f  of  j\  of  3|. 

124f..  If  I  of  /.  of  S  '^  P«iy'fo«'i  off  of,j9^of  iOO  acres, 
.  iiOW  much  can  be  bnu«?hi  lor  '?-  of  X  of  123*4.^/.  N.  8rofia  ? 


^54:  Questions, 

1^45.  If  y\  ot  25  Eng.  guineas  pay  for  a.  of  |  of  f  of 
556-34  yds.,  what  will  f  of  f  of  |  of  325^  13s.  6(1/N.  York, 
pay  for  ? 

1246.  If  f  of  1  of  I  of  235  guilders  pay  for  |  of  |  of  J 
of  39fi|i  yds.,  how  many  dolls,  will  pav  for  f  of  -/^  of  |  of 
of  23-678  yds.  ? 

1247.  If  I  of  1  of  3  lb.  of  tea,  serve  9  persons  f  of  |  of 
91  months,  how  many  persons  will  J  of  i  of  ^  of  10^  lb. 
serve  ^  of  J  of  15  months  ? 

1248.  If  34'56L  ster.  be  the  interest  of  356  Eng.  guineas, 
for  1  of  I  of  345-67  days,  how  much  sterling  will  be  the  in* 
terest  of  3867f  dolls,  for  i  of  f  of  4  of  234^4  days  ? 

1249.  Divide  3^  by  ^  of  |-"of  |  of  J  of>f,  and  tell  the 
difference  between  the  square  and  cube  of  the  quotient. 

1 250.  If  the  walls  of  the  temple  of  Solomon,  had  been  5 
feet  thick  when  finished,  and  one  inch  of  the  outside  and 
one  inch  of  the  inside  had  been  fine  gold,  how  many  talents 
would  it  have  required,  allowing  one  foot  high  on  the  inside 
to  have  been  occupied  by  the  floor  ?  (See  No.  399.) 

1251.  Ascending  bodies  ar-e  retarded  in  Ihe  same  ratio 
hat  descending  bodies  are  accelerated  ;    therefore,  if  a  ball 

discharged  from  a  gun  perpendicularly  into  the  air,  returned 
lo  the  earth  in  10  seconds,  how  high  did  it  ascend  ? 

l'^52.  Find  the  number  of  solid  feet  in  a  load  of  wood, 
which  is  4  ft.  6  in.  high,  3  ft  10  in.  wide,  and  9  ft.  4»  in.  long. 

1253.  A  laborer  was  hired  for  50  days,  upon  condition  that 
lor  every  day  he  labored  he  should  receive  8s.,  and  for  every 
day  he  was  idle  he  should  forfeit  3s.  At  the  end  of  the  time, 
lie  received  9/.  lis. ;  how' many  days  did  he  labor  ? 

1254.  Tell  the  area  of  a  circle  whose  radius  is  14  chains. 

1255.  Find  the  difference  in  th6  depth  of  two  wells,  into 
which  a  bullet  let  fall,  reaches  the  bottom  in  4  seconds  and 
6  seconds  respectively. 

1256.  If  I  of  I  of  I  of  10^  lb.  of  tea  serve  5  persons  3| 
months,  how  much  will  serve  7  persons  -|  off  of  |  of  ^  mo.  ? 

1257.  Tell  the  area  of  a  circle  whose  circumference  is  25 
chains. 

1258.  What  mu>^t  be  paid  for  a  perpetuity  of  100  dollars, 
to  commence  in  4  years,  discount  at  4^  per  cent  compound 
interest  ? 

12  .9.  If  a  man  travels  34-56  miles  in  4|  days,  when  the 
days  are  13f  hours  long ;  how  far  can  he  travel  in  10-45  days^, 
when  they  are  15'37  hours  long  ? 


QuesVoiTS.  235 

1260.  Tell  the  difference  between  SO  square  rods,  and  30 
:ods  square. 

126i.  In  what  time  will  90Ci.  amount  to  23U.  10s.  6d.  at 
5  per  cent  compound  interest  ? 

1£62.  Tell  the  superficial  content  of  a  pyramid,  the  base 
of  which  is  an  equilateral  triangle,  each  side  measunng  13 
feet,  and  the  slant  height  35  feet. 

1263.  Find  the  solid  content  of  a  cone,  the  diameter  of 
vhose  base  is  17  feet,  and  the  perpendicular  height  35  fpet. 

1264.  What  is  the  circumference  of  a  circle,  whose  dia* 
meter  is  36  rods  ? 

126,5.  Multiply  i  of  |  of  the  square  of  45,  by  f  of  |^  of  the 
square  root  of  144. 

1266.  What  number  is  that,  which  being  multiplied  by  18, 
and  the  product  divided  by  12,  the  quotient  is  30o  ? 

1267.  Add  f  of  f  of  the  square  of  ||4,  to  ^  of  |  of  |  of  the 
sq u a r e  roo t  of  ^ ^V* 

1268.  What  number  is  that,  wliich  being  increased  by  J, 
■ ,  \,  and  ^  of  J  of  itself,  and  the  sum  divided  by  20,  the  quo- 
ient  will  be  77? 

1269.  Add  the  square  of  ||  to  the  cube  of  f|-,  and  tell 
what  is  i  of  I  of  f  of  the  sum. 

1270.  Tell  the  solid  content  of  a  cone,  the  circumference 
of  whose  base  is  45  feet,  and  its  perpendicular  height  34  feet. 

1271.  Find  the  difference  between  the  simple  interest  of 
375  dolls,  for  6  year*,  at  7  per  cent,  and  the  discount  of  the 
same  sum,  at  the  same  rate  and  time. 

1272.  Add  the  cube  of  ||  to  the  square  of  |f,  and  tell 
what  is  ^  of  ^  of  its  square  root. 

1573.  AVhich  is  the  most  valuable,  and  how  much  so.  an 
annuity  of  500  dolls,  for  8  years,  or  a  perpetuity  of  500  dolls, 
to  commence  after  8  years,  discounting  at  6  per  centcom-r 
pound  interest  ? 

1274.  From  ^t  ^ake  4  of  ^  of  itself,  and  tell  what  is  the 
Bum  of  the  square  and  cube  of  the  remainder. 

1275.  At  what  rate  per  cent,  compound  interest,  will  200 
dolls,  amount  to  S262-4*;5392,  in  4  years  ? 

1 276.  A  man  sold  |  of  his  sheep,  and  12  more,  at  one  time ; 
And  at  another,  J  of  the  remainder,  and  i5  wiore,  and  had 
137  left ;  how  many  had  he  at  first  ? 

1277.  Find  the  content  of  a  circular  ring  2  rods  broad, 
round  a  circulai*  fish  pond  of  3  acres  ? 


236  Questions. 

1978.  A  person  who  owned  |  of  a  ship,  sold  f  of  his  share 
for  500  do  is. ;  what  was  the  ship  worth  ? 

1 2.  9.  In  an  orchard,  |- are  apple  trees,  ^  pear  trees,  ^ 
cherry  trees,  i  peach  trees,  and  lO  plum  trees;  how  many 
trees  in  all  ? 

12H).  What  is  the  content  of  a  circular  ring  3  rods  broad, 
the  inner  circun^ference  of  which  measures  346  rods  ? 

1281.  If  §  of  i  of  I  of  a  ship  is  worth  -}  off  of  |  of  J  of  her 
cargo,  that  part  of  the  cargo  being  worth  1 200/. ;  what  is  the 
value  of  the  ship  and  cargo  together  •' 

\^S^.  Tell  the  amount  of  an  annuity  of  650  dollars,  for  7 
years,  at  6  per  cent,  simple  interest  ? 

12s3.  A,  B.  and  C,  purchased  a  vessel  in  company;  A 
paid  |,  B  |,  and  C  loo  dolls  ;  what  was  the  whole  ? 

1284.  tell  the  cube  root  of  i  of  f  of  |  of  8. 

1285.  Which  is  the  larger,  and  how  much  so,  a  square  of 
450  rods  circumference,  or  a  circle  of  the  same  circumfe- 
rence ? 

1285.  What  sum  of  money  will  produce  as  much  interest 
in  4^  years,  as  345  dolls,  would  in  7§  years  ? 

li;87.  Tell  the  cube  root  of  ^of  i  of  |  of  21. 

1 2^-;8.   What  part  of  2s.  6d.  is  |-  of  f  of  |  of  Is.  6d.  ? 

1  2  89.  if  a  cone  whose  slant  height  is  20  feet,  contains  2 1  6 
solid  feet,  how  far  from  the  base  on  the  slant  height,  must  it 
be  cut,  by  a  section  parallel  to  the  base,  to  be  divided  into 
two  equal  parts  ? 

1290.  Find  the  cube  root  of  the  sum  of  the  square  roots  of 
529  and  1936. 

129  .  What  number  is  that,  from  which,  iff  of  |  is  taken, 
the  remainder  is  ^  of|? 

1  ::92.  Tell  the  superficial  content  of  a  globe  which  is  3 
feet  in  diameter. 

1293.  Teil  the  cube  root  of  the  diiference  of  the  square 
roots  of  3 1 66  and  5776. 

1294.  What  number  is  that,  to  which,  iff  of|^of|ff  be 
added,  the  sum  will  be  1  ? 

I  v95.  What  number  is  that,  which  being  multiplied  by  | 
off  of  2,  the  product  will  be  1  ? 

r29fv.  A,  B,.and  C,  bought  a  ship,  which  cost  2500  dolls. 
B  paid  lOt;  dolls,  more  than  A  ;  and  C,  iOO  dolls,  more  than 
A  and  B  both.  They  furnished  a  cargo  which  cost  250  dolls. 
less  than  twice  the  vaiue  of  the  ship  ;  and  the  expense  of 
fitting  out  the  vessel  was  |  of  f  of  |  of  j\  of  the  value  of  both 


%iestions.  237 

ship  and  cargo-  The  profits  of  the  voyage  were  25  per  cent 
on  the  whole,  which  were  to  be  shared  according  to  the  in- 
terest ot*  each  in  the  ship :  what  was  A's  share  of  the  gain  ? 

5  297.  A  man  received  for  his  wages,  one  graki  of  wheat 
for  the  fir  t  day,  4  for  the  second,  6  for  the  third, and  so  on; 
what  is  the  amount  of  30  days'  labor,  if  the  wheat  is  worth 
1  doll,  a  bushel,  and  7680  grains  of  wheat  make  a  pint  ? 

129^,  What  number  is  that,  from  one  half  of  the  square- of 
which,  if  I,  |,  |,  f,  and  ^  of  the  number,  and  123  more,  be 
subtracted,  the  square  root  of  the  remainder  is  294  ? 

1299.  Two  merchants,  A  and  b,  began  trade  with  the 
same  capital.  A  was  successful,  and  gained  f  off  of  |^  as 
much  as  his  stock  ;  but  B  lost  |  of  f  of  ^  of  ^^AO  dolls,  and 
20  dolls,  more ;  and  then  their  capitals  were  in  the  propor- 
tion of  5  to  3  :  what  capital  did  each  begin  with  ? 

1300.  A,  B,  and  C,  joined  stock  n  trade,  and  made  up 
a  capital  often  thousand  dollars.  B  furnished  ^25  dolls,  more 
than  A ;  and  C,  600  dolls,  less  than  A  and  B  together.  At 
the  end  of  6  months,  A  took  out  his  stock;  and  at  the  end  of 
10  months,  B  took  out  his.  At  the  year's  end,  the  gain  was 
a  sum  equal  to  |  of  y^^  of  |^  of  3  times  the  stock  of  C  :  what 
was  each  man's  share  of  the  gain  ? 

1301.  A  and  B  bought  300  acres  of  land  for  600  dollars,  of 
wliich  they  paid  equal  sums.  One  part  of  the  lot  prov^ig  better 
than  the  other,  A  says  to  B,  if  you  will  let  me  have  my  choice, 
your  land  shall  cost  you  75  cents  an  acre  less  than  mine. 
If  B  agrees  to  this  proposal,  how  much  land  will  he  have  ? 

1302.  1  wish  to  fence  a  circular  piece  of  ground,  with  rails 
which  shall  be  so  long  as  to  make  '  0  feet  to  each  length,  and 
the  fence  to  be  5  rails  high,  and  to  have  as  many  acres  of 
ground  as  1  have  rails ;  how  many  acres  will  there  be  ? 

1303.  Suppose  the  frustum  of  a  right  pyramid  to  be  4  feet 
square  at  the  base,  and  one  foot  square  at  the  top,  and  the 
slant  height  .0  feet ;  and  a  rope  -2  inches  thick,  to  be  wound 
round  it,  so  as  to  cover  its  sides  from  the  bottom  to  the  top ; 
how  long  is  the  rope  ? 

1304.  A,  B,  C,  and  D,  purchased  a  grindstone  in  compa- 
ny ;  and  the  sums  they  paid  respectively,  were  such  that  A, 
B,  and  C  together,  paid  3  dolls.  9  cents ;  A,  B,  and  D  to- 
gether, 1/.  '3s.  3-36d.  N.  York  cur;  B,  C,  and  D  together, 
ll,  .s.  li-52d.  N.  England  cur.;  and  A,  C,  and  D  together, 
M.  i  s.  3d.  sterling.  The  grindstone  was  3  feet  in  diameter, 
and  5  inches  thick ;  and  there  was  a  hole  in  the  middle,  4 


£38  ^uestians, 

inches  in  diameter.  A  is  to  have  it  first,  and  grind  off  in 
proportion  to  the  sum  he  paid ;  then  B,  C,  and  D  respectively^ 
I  demand  the  breadth  of  the  circular  ring  which  each  is  to 
grind  off. 

1305.  A  party  of  17  persons  wish  to  go  to  a  place  25  miles 
distant.  They  have  but  one  carriage,  which  will  carry  5 
persons.  At  8  o'clock  in  the  forenoon,  the  carriage  sets  out 
with  5  of  the  party,  and  goes  on  at  the  rate  of  6  miles  an  hour, 
till  half  past  9,  when  it  stops  1 5  minutes.  Finding  the  roads 
worse,  it  then  goes  at  the  rate  of  5i  miles  an  hour,  till  a 
quarter  past  10,  and  then  5  miles  an  hour  to  the  end,  having 
made  one  more  stop  of  20  minutes.  At  the  end,  it  stops  18 
minutes,  and  then  goes  back,  with  one  person  to  drive,  at  the 
rate  of  7  miles  an  hour.  The  rest  of  the  party  set  out  on 
foot,  10  minutes  after  the  carriage,  and  w^alk  at  the  rate  of  4 
m:les  an  hour,  till  a  quarter  before  10,  when  they  stop  20 
minutes.  After  this,  they  walk  on  at  the  rate  of  Sf  miles  an 
hour,  till  half  past  10,  when  they  stop  15  minutes.  They 
then  walk  on  at  the  rate  of  3  miles  an  hour,  stopping  once 
more  15  minutes,  till  they  meet  the  carriage.  The  carriage 
takes  up  4.  of  them,  and  goes  on  to  the  end,  at  the  rate  of  5 
miles  an  hour.  After  resting  15  minutes,  it  goes  bnck  again, 
at  the  rate  of  6  miles  an  hour.  The  remainirig  8,  after  resting 
20  minutes,  walk  on  at  the  rate  of  2^  miles  an  hour,  till  they 
meet  the  carriage  again.  Then  the  carriage  takes  up  4  more, 
and  carries  them  to  the  end,  at  the  rate  of  5  miles  an  hour, 
and  returns  without  stopping,  at  the  rate  of  r>  miles  an  hour. 
The  remaining  4  walk  on  at  the  rate  of  2^  miles  an  hour,  1 11 
they  meet  the  carriage,  and  then  ride  to  the  end,  at  the  rate 
of  5  miles  an  hour.  How  far  does  the  carnage  travel  in  all, 
and  at  what  o'clock  do  the  last  of  the  party  reach  the  place 
of  destination  ? 

306.  Suppose  A,  B,  and  C,  can  do  apiece  of  work  in  165 
days  ;  B,  C,  and  D,  in  220  days ;  A,  B,  and  D,  in  18  days: 
A,  C,  and  D,  in  198  days  :  how  long  will  it  take  each  one  to 
do  it  separately ;  and  how  long,  if  they  all  work  together  ? 

1307.  Divide  15  into  2  such  parts,  that  their  product  shall 

Ha     7  4  8  4 

130^.  If  6?i  bushels  of  oats  aref  sufficient  for  12  horses  for 
4  weeks,  and  236^  bushels  are  sufficient  for  21  horses  for  9 
weeks ;  how  many  horses  will  627^  bushels  suffice  for  I S 
weeks,  proceeding  in  the  same  ratio  i* 


Questions*  ^39 

1309.  A  and  "B  join  in  trade,  and  make  up  a  capital  of 
such  a  number  of  dollars,  that  if  it  were  diminished  b^  ^  of 
J  of  f  of  itself,  and  then  by  |  of  |  of  f  of  the  remainder, 
there  would  be  648  dolls,  left.  Their  gain  was  50  per  cent 
on  their  capital,  and  is  to  be  divided  in  proportion  to  their 
shares  of  the  capital.  A's  share  is  to  be  the  most,  and  to  be 
such  a  sum,  that  if  multiplied  by  B's,  and  that  product  mul- 
tiplied by  the  whole  capital,  and  that  product  divided  by  ^  of 
I  off  off  of  the  capital,  the  quotient  would  be  324000  dols. 
What  is  each  man's  share  of  the  gain  ? 

1310.  Six  persons,  to  amuse  themselves,  threw  upon  the 
ground  a  sum  of  money,  to  see  how  much  each  could  pick  up ; 
and  the  first  time,  they  gathered  as  follows  :  A,  |  off  of  f  of 
a  pound  sterling ;  B,  f  of  |  as  much,  and  |  of  a  dollar  be- 
sides ;  C,  I  of  §  of  I  as  much  as  B,  and  ^^  of  a  penny,  and  f 
of  7  livres  besides  ;  D,  f  of  |-  of  f  as  much  as  C,  and  8d.  and 
^  of  a  farthing,  and  J  of  |  of  4  rubles  besides  ;  E,  |  of  f  of 
i  as  much  as  D,  and  iff  of  a  penny,  and  f  of  f  of  5i  crowns 
besides  ;  and  F,  the  rest,  which  was  |  of  |  of  |  of  |  as  much 
as  E,  and  J  of  J  of  |  of  |  of  2|  guilders  besides. 

Then  the  money  was  thrown  down  again,  and  the  second 
time  they  gathered  as  follows  :  A,  |  of  4  of  f  of  |  of  what  he 
had  the  first  time ;  B,  f  of  f  of  -^  of  ^  of  J  as  much  as  A  ;  C, 
I  of  I  of  f  of  45  times  as  much  as  B  ;  D,  f  of  f  of  ^^^  of  |  of 
J  of  1 92  times  as  much  as  C  ;  E,  |  of  -/^  of  |  of  |  of  f  of  25 
times  as  much  as  D ;  and  F,  the  rest. 

Now,  as  F  had  so  much  the  most,  he  threw  down  his  again ; 
and  A  got  |  of  f  of  ^  of  |  of  it,  and  \  shilling,  and  g^r^  of  a 
shilling  besides  ;  B,  |  of  |^  of  f  of  J  of  12  times  as  much  as 
A  ;  C,  1^  of  ^  of  ^  of  J  of  f  of  35  times  as  much  as  B  ;  D,  |^  of 
^  of  I  of  j\  of  i  60  times  as  much  as  C  ;  F,  |  of  |  of  ^  of  -J  of 
24  times  as  much  as  D  ;  and  E,  the  rest. 

Then  E  gave  to  B  |  of  a  marc  banco,  to  C  3  livres,  and  to 
¥  half  a  crown  ;  and  A  gave  to  B  §  of  a  rupee,  and  to  F  one 
rial  of  plate. 

How  much,  in  sterling  money,  was  the  sum  thrown  down  ; 
and  how  much  had  each  person  at  last  ? 


MJSrn  OF  FART  JL 


Page    26,  line    3j  for  31  read  32. 

32,     ,,   27    right  hand  c.lumn,  insert  9  under  8. 
;,      43,     ,,    15    after    i?is.  insert  D. 
„      48,     „    55,  *or  112,  40.  read  112,  46 
;,      49,     „    38,  for  6  s»  venths,  rea'l  5  sevenths, 

39,  for  5  nitiths,  r-  ad  6  ninths 
.,      60,     ,,    14,  for  fraction,  read  fractions, 
20,  ff.r  94^1  read  4_i 

J,  G8,  „    31,  read  64  and  13,  11  at.d  17,  49. 

,,  70,  „    34    right  band  colun.n,  reau  80 

„  79,  „    28,  f  .r  14  A.  read  1-4/.. 

„  95,  ,,    40,  for  Z03  r  ad  Xl03 

„  115,  „    18    undei  30,  read  900 

„  134,  „    26,  for  ^9461-  &c  ,  read  ^6461-  &c. 

„  144,  „    25,  lor  bame,  <  tad  area. 

.,  145,  „      6,  rt.ad.  20-92!  875  acres. 

„  149,  „    34,  for  7496,  read  17496. 

„  150.  ,   23ik24,  •bi-12i«28,  tead-12&'28, 

,,  151,  „    '17,  for  1  1624-  read  1  162— 

5,  154,  „      9,  r  ad    5  own,  20 

2il,  read,   Clarkson,  150, 

40,  r^ad    Ann  ricn.  29815000. 
V,  155,  „    13,  read    rear  1799 

14,  read  i^s';e(i  2967000. 
->    156,     „      2,  r  ad   v.  ar  1782. 

29,  read.  jClSOOO. 

43,  read    8758  8. 
,,    160,     „    13,  for  2700   read,  27000. 
„    168,    „    38,  read,  of  2t7  3. 

41,  for  A.    ead  ih 

,5    177,     „    31,  for  22095  r(  ad,  220959. 

and  fcr  4065119,  read  40651L 
j5    188,     „      4,  for  1377000   read  1377000. 

36,  for  ^52049,  read  232049. 
„    190,     „      6,  for  were,  r^  ad  was. 

„    17.  for  30,  read  80. 

„    25,  for  245,  read  285 

„    42,  for  44187,  read  44487. 
5,   205,     „   25,  for  29,  read  19. 
.,    211,     „    33,  read  4036658 
„    2>3,     „     7,  f  r  335  '-4,  read^65  1-4 
5,    224,    „   31,  veaa  2000. 


*)  f\ 


